diff --git a/previews/PR98/404.html b/previews/PR98/404.html index 93878e7..405d68b 100644 --- a/previews/PR98/404.html +++ b/previews/PR98/404.html @@ -9,7 +9,7 @@ - + @@ -23,7 +23,7 @@
- + \ No newline at end of file diff --git a/previews/PR98/api/basis.html b/previews/PR98/api/basis.html index 8d782b4..7f3b038 100644 --- a/previews/PR98/api/basis.html +++ b/previews/PR98/api/basis.html @@ -9,11 +9,11 @@ - + - + - + @@ -25,8 +25,8 @@ -
Boltz.jl Docs

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Boltz.Basis API Reference ​

Warning

The function calls for these basis functions should be considered experimental and are subject to change without deprecation. However, the functions themselves are stable and can be freely used in combination with the other Layers and Models.

Boltz.Basis.Chebyshev Method
julia
Chebyshev(n; dim::Int=1)

Constructs a Chebyshev basis of the form [T0(x),T1(x),…,Tn−1(x)] where Tj(.) is the jth Chebyshev polynomial of the first kind.

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Cos Method
julia
Cos(n; dim::Int=1)

Constructs a cosine basis of the form [cos⁡(x),cos⁡(2x),…,cos⁡(nx)].

Arguments

  • n: number of terms in the cosine expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Fourier Method
julia
Fourier(n; dim=1)

Constructs a Fourier basis of the form

Fj(x)={cos(j2x)if j is evensin(j2x)if j is odd

Arguments

  • n: number of terms in the Fourier expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Legendre Method
julia
Legendre(n; dim::Int=1)

Constructs a Legendre basis of the form [P0(x),P1(x),…,Pn−1(x)] where Pj(.) is the jth Legendre polynomial.

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Polynomial Method
julia
Polynomial(n; dim::Int=1)

Constructs a Polynomial basis of the form [1,x,…,x(n−1)].

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Sin Method
julia
Sin(n; dim::Int=1)

Constructs a sine basis of the form [sin⁡(x),sin⁡(2x),…,sin⁡(nx)].

Arguments

  • n: number of terms in the sine expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

- +
Boltz.jl Docs

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Boltz.Basis API Reference ​

Warning

The function calls for these basis functions should be considered experimental and are subject to change without deprecation. However, the functions themselves are stable and can be freely used in combination with the other Layers and Models.

Boltz.Basis.Chebyshev Method
julia
Chebyshev(n; dim::Int=1)

Constructs a Chebyshev basis of the form [T0(x),T1(x),…,Tn−1(x)] where Tj(.) is the jth Chebyshev polynomial of the first kind.

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Cos Method
julia
Cos(n; dim::Int=1)

Constructs a cosine basis of the form [cos⁡(x),cos⁡(2x),…,cos⁡(nx)].

Arguments

  • n: number of terms in the cosine expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Fourier Method
julia
Fourier(n; dim=1)

Constructs a Fourier basis of the form

Fj(x)={cos(j2x)if j is evensin(j2x)if j is odd

Arguments

  • n: number of terms in the Fourier expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Legendre Method
julia
Legendre(n; dim::Int=1)

Constructs a Legendre basis of the form [P0(x),P1(x),…,Pn−1(x)] where Pj(.) is the jth Legendre polynomial.

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Polynomial Method
julia
Polynomial(n; dim::Int=1)

Constructs a Polynomial basis of the form [1,x,…,x(n−1)].

Arguments

  • n: number of terms in the polynomial expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

Boltz.Basis.Sin Method
julia
Sin(n; dim::Int=1)

Constructs a sine basis of the form [sin⁡(x),sin⁡(2x),…,sin⁡(nx)].

Arguments

  • n: number of terms in the sine expansion.

Keyword Arguments

  • dim::Int=1: The dimension along which the basis functions are applied.

source

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Boltz.jl Docs

Appearance

- + \ No newline at end of file diff --git a/previews/PR98/api/layers.html b/previews/PR98/api/layers.html index 2e59e88..89b14db 100644 --- a/previews/PR98/api/layers.html +++ b/previews/PR98/api/layers.html @@ -9,11 +9,11 @@ - + - + - + @@ -25,9 +25,9 @@ -
Boltz.jl Docs

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Boltz.Layers API Reference ​

Boltz.Layers.ClassTokens Type
julia
ClassTokens(dim; init=zeros32)

Appends class tokens to an input with embedding dimension dim for use in many vision transformer models.

source

Boltz.Layers.ConvNormActivation Type
julia
ConvNormActivation(kernel_size::Dims, in_chs::Integer, hidden_chs::Dims{N},
+    
Boltz.jl Docs
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Boltz.Layers API Reference ​
Boltz.Layers.ClassTokens Type
julia
ClassTokens(dim; init=zeros32)
Appends class tokens to an input with embedding dimension dim for use in many vision transformer models.
source
Boltz.Layers.ConvNormActivation Type
julia
ConvNormActivation(kernel_size::Dims, in_chs::Integer, hidden_chs::Dims{N},
     activation; norm_layer=nothing, conv_kwargs=(;), norm_kwargs=(;),
-    last_layer_activation::Bool=false) where {N}
Construct a Chain of convolutional layers with normalization and activation functions.
Arguments
  • kernel_size: size of the convolutional kernel
  • in_chs: number of input channels
  • hidden_chs: dimensions of the hidden layers
  • activation: activation function
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • conv_kwargs: keyword arguments for the convolutional layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
Boltz.Layers.DynamicExpressionsLayer Type
julia
DynamicExpressionsLayer(operator_enum::OperatorEnum, expressions::Node...;
+    last_layer_activation::Bool=false) where {N}
Construct a Chain of convolutional layers with normalization and activation functions.
Arguments
  • kernel_size: size of the convolutional kernel
  • in_chs: number of input channels
  • hidden_chs: dimensions of the hidden layers
  • activation: activation function
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • conv_kwargs: keyword arguments for the convolutional layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
Boltz.Layers.DynamicExpressionsLayer Type
julia
DynamicExpressionsLayer(operator_enum::OperatorEnum, expressions::Node...;
     eval_options::EvalOptions=EvalOptions())
 DynamicExpressionsLayer(operator_enum::OperatorEnum,
     expressions::AbstractVector{<:Node}; kwargs...)
Wraps a DynamicExpressions.jl Node into a Lux layer and allows the constant nodes to be updated using any of the AD Backends.
For details about these expressions, refer to the DynamicExpressions.jl documentation.
Arguments
  • operator_enum: OperatorEnum from DynamicExpressions.jl
  • expressions: Node from DynamicExpressions.jl or AbstractVector{<:Node}
Keyword Arguments
  • turbo: Use LoopVectorization.jl for faster evaluation (Deprecated)
  • bumper: Use Bumper.jl for faster evaluation (Deprecated)
  • eval_options: EvalOptions from DynamicExpressions.jl
These options are simply forwarded to DynamicExpressions.jl's eval_tree_array and eval_grad_tree_array function.
Extended Help
Example
julia
julia> operators = OperatorEnum(; binary_operators=[+, -, *], unary_operators=[cos]);
@@ -65,23 +65,23 @@
 true
 
 julia> ∂ps.layer_1.layer_2.params ≈ Float32[-31.0, 90.0]
-true
source
Boltz.Layers.HamiltonianNN Type
julia
HamiltonianNN{FST}(model; autodiff=nothing) where {FST}
Constructs a Hamiltonian Neural Network (Greydanus et al., 2019). This neural network is useful for learning symmetries and conservation laws by supervision on the gradients of the trajectories. It takes as input a concatenated vector of length 2n containing the position (of size n) and momentum (of size n) of the particles. It then returns the time derivatives for position and momentum.
Arguments
  • FST: If true, then the type of the state returned by the model must be same as the type of the input state. See the documentation on StatefulLuxLayer for more information.
  • model: A Lux.AbstractLuxLayer neural network that returns the Hamiltonian of the system. The model must return a "batched scalar", i.e. all the dimensions of the output except the last one must be equal to 1. The last dimension must be equal to the batchsize of the input.
Keyword Arguments
  • autodiff: The autodiff framework to be used for the internal Hamiltonian computation. The default is nothing, which selects the best possible backend available. The available options are AutoForwardDiff and AutoZygote.
Autodiff Backends
autodiffPackage NeededNotes
AutoZygoteZygote.jlPreferred Backend. Chosen if Zygote is loaded and autodiff is nothing.
AutoForwardDiffChosen if Zygote is not loaded and autodiff is nothing.
Note
This layer uses nested autodiff. Please refer to the manual entry on Nested Autodiff for more information and known limitations.
source
Boltz.Layers.MLP Type
julia
MLP(in_dims::Integer, hidden_dims::Dims{N}, activation=NNlib.relu; norm_layer=nothing,
+true
source
Boltz.Layers.HamiltonianNN Type
julia
HamiltonianNN{FST}(model; autodiff=nothing) where {FST}
Constructs a Hamiltonian Neural Network (Greydanus et al., 2019). This neural network is useful for learning symmetries and conservation laws by supervision on the gradients of the trajectories. It takes as input a concatenated vector of length 2n containing the position (of size n) and momentum (of size n) of the particles. It then returns the time derivatives for position and momentum.
Arguments
  • FST: If true, then the type of the state returned by the model must be same as the type of the input state. See the documentation on StatefulLuxLayer for more information.
  • model: A Lux.AbstractLuxLayer neural network that returns the Hamiltonian of the system. The model must return a "batched scalar", i.e. all the dimensions of the output except the last one must be equal to 1. The last dimension must be equal to the batchsize of the input.
Keyword Arguments
  • autodiff: The autodiff framework to be used for the internal Hamiltonian computation. The default is nothing, which selects the best possible backend available. The available options are AutoForwardDiff and AutoZygote.
Autodiff Backends
autodiffPackage NeededNotes
AutoZygoteZygote.jlPreferred Backend. Chosen if Zygote is loaded and autodiff is nothing.
AutoForwardDiffChosen if Zygote is not loaded and autodiff is nothing.
Note
This layer uses nested autodiff. Please refer to the manual entry on Nested Autodiff for more information and known limitations.
source
Boltz.Layers.MLP Type
julia
MLP(in_dims::Integer, hidden_dims::Dims{N}, activation=NNlib.relu; norm_layer=nothing,
     dropout_rate::Real=0.0f0, dense_kwargs=(;), norm_kwargs=(;),
-    last_layer_activation=false) where {N}
Construct a multi-layer perceptron (MLP) with dense layers, optional normalization layers, and dropout.
Arguments
  • in_dims: number of input dimensions
  • hidden_dims: dimensions of the hidden layers
  • activation: activation function (stacked after the normalization layer, if present else after the dense layer)
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • dropout_rate: dropout rate (default: 0.0f0)
  • dense_kwargs: keyword arguments for the dense layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
Boltz.Layers.MultiHeadSelfAttention Type
julia
MultiHeadSelfAttention(in_planes::Int, number_heads::Int; use_qkv_bias::Bool=false,
-    attention_dropout_rate::T=0.0f0, projection_dropout_rate::T=0.0f0)
Multi-head self-attention layer
Arguments
  • planes: number of input channels
  • nheads: number of heads
  • use_qkv_bias: whether to use bias in the layer to get the query, key and value
  • attn_dropout_prob: dropout probability after the self-attention layer
  • proj_dropout_prob: dropout probability after the projection layer
source
Boltz.Layers.PatchEmbedding Type
julia
PatchEmbedding(image_size, patch_size, in_channels, embed_planes;
-    norm_layer=Returns(Lux.NoOpLayer()), flatten=true)
Constructs a patch embedding layer with the given image size, patch size, input channels, and embedding planes. The patch size must be a divisor of the image size.
Arguments
  • image_size: image size as a tuple
  • patch_size: patch size as a tuple
  • in_channels: number of input channels
  • embed_planes: number of embedding planes
Keyword Arguments
  • norm_layer: Takes the embedding planes as input and returns a layer that normalizes the embedding planes. Defaults to no normalization.
  • flatten: set to true to flatten the output of the convolutional layer
source
Boltz.Layers.PeriodicEmbedding Type
julia
PeriodicEmbedding(idxs, periods)
Create an embedding periodic in some inputs with specified periods. Input indices not in idxs are passed through unchanged, but inputs in idxs are moved to the end of the output and replaced with their sines, followed by their cosines (scaled appropriately to have the specified periods). This smooth embedding preserves phase information and enforces periodicity.
For example, layer = PeriodicEmbedding([2, 3], [3.0, 1.0]) will create a layer periodic in the second input with period 3.0 and periodic in the third input with period 1.0. In this case, layer([a, b, c, d], st) == ([a, d, sinpi(2 / 3.0 * b), sinpi(2 / 1.0 * c), cospi(2 / 3.0 * b), cospi(2 / 1.0 * c)], st).
Arguments
  • idxs: Indices of the periodic inputs
  • periods: Periods of the periodic inputs, in the same order as in idxs
Inputs
  • x must be an AbstractArray with issubset(idxs, axes(x, 1))
  • st must be a NamedTuple where st.k = 2 ./ periods, but on the same device as x
Returns
  • AbstractArray of size (size(x, 1) + length(idxs), ...) where ... are the other dimensions of x.
  • st, unchanged
Example
julia
julia> layer = Layers.PeriodicEmbedding([2], [4.0])
+    last_layer_activation=false) where {N}
Construct a multi-layer perceptron (MLP) with dense layers, optional normalization layers, and dropout.
Arguments
  • in_dims: number of input dimensions
  • hidden_dims: dimensions of the hidden layers
  • activation: activation function (stacked after the normalization layer, if present else after the dense layer)
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • dropout_rate: dropout rate (default: 0.0f0)
  • dense_kwargs: keyword arguments for the dense layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
Boltz.Layers.MultiHeadSelfAttention Type
julia
MultiHeadSelfAttention(in_planes::Int, number_heads::Int; use_qkv_bias::Bool=false,
+    attention_dropout_rate::T=0.0f0, projection_dropout_rate::T=0.0f0)
Multi-head self-attention layer
Arguments
  • planes: number of input channels
  • nheads: number of heads
  • use_qkv_bias: whether to use bias in the layer to get the query, key and value
  • attn_dropout_prob: dropout probability after the self-attention layer
  • proj_dropout_prob: dropout probability after the projection layer
source
Boltz.Layers.PatchEmbedding Type
julia
PatchEmbedding(image_size, patch_size, in_channels, embed_planes;
+    norm_layer=Returns(Lux.NoOpLayer()), flatten=true)
Constructs a patch embedding layer with the given image size, patch size, input channels, and embedding planes. The patch size must be a divisor of the image size.
Arguments
  • image_size: image size as a tuple
  • patch_size: patch size as a tuple
  • in_channels: number of input channels
  • embed_planes: number of embedding planes
Keyword Arguments
  • norm_layer: Takes the embedding planes as input and returns a layer that normalizes the embedding planes. Defaults to no normalization.
  • flatten: set to true to flatten the output of the convolutional layer
source
Boltz.Layers.PeriodicEmbedding Type
julia
PeriodicEmbedding(idxs, periods)
Create an embedding periodic in some inputs with specified periods. Input indices not in idxs are passed through unchanged, but inputs in idxs are moved to the end of the output and replaced with their sines, followed by their cosines (scaled appropriately to have the specified periods). This smooth embedding preserves phase information and enforces periodicity.
For example, layer = PeriodicEmbedding([2, 3], [3.0, 1.0]) will create a layer periodic in the second input with period 3.0 and periodic in the third input with period 1.0. In this case, layer([a, b, c, d], st) == ([a, d, sinpi(2 / 3.0 * b), sinpi(2 / 1.0 * c), cospi(2 / 3.0 * b), cospi(2 / 1.0 * c)], st).
Arguments
  • idxs: Indices of the periodic inputs
  • periods: Periods of the periodic inputs, in the same order as in idxs
Inputs
  • x must be an AbstractArray with issubset(idxs, axes(x, 1))
  • st must be a NamedTuple where st.k = 2 ./ periods, but on the same device as x
Returns
  • AbstractArray of size (size(x, 1) + length(idxs), ...) where ... are the other dimensions of x.
  • st, unchanged
Example
julia
julia> layer = Layers.PeriodicEmbedding([2], [4.0])
 PeriodicEmbedding([2], [4.0])
 
 julia> ps, st = Lux.setup(Random.default_rng(), layer);
 
 julia> all(layer([1.1, 2.2, 3.3], ps, st)[1] .==
            [1.1, 3.3, sinpi(2 / 4.0 * 2.2), cospi(2 / 4.0 * 2.2)])
-true
source
Boltz.Layers.SplineLayer Type
julia
SplineLayer(in_dims, grid_min, grid_max, grid_step, basis::Type{Basis};
-    train_grid::Union{Val, Bool}=Val(false), init_saved_points=nothing)
Constructs a spline layer with the given basis function.
Arguments
  • in_dims: input dimensions of the layer. This must be a tuple of integers, to construct a flat vector of saved_points pass in ().
  • grid_min: minimum value of the grid.
  • grid_max: maximum value of the grid.
  • grid_step: step size of the grid.
  • basis: basis function to use for the interpolation. Currently only the basis functions from DataInterpolations.jl are supported:
    1. ConstantInterpolation
    2. LinearInterpolation
    3. QuadraticInterpolation
    4. QuadraticSpline
    5. CubicSpline
Keyword Arguments
  • train_grid: whether to train the grid or not.
  • init_saved_points: values of the function at multiples of the time step. Initialized by default to a random vector sampled from the unit normal. Alternatively, can take a function with the signature init_saved_points(rng, in_dims, grid_min, grid_max, grid_step).
Warning
Currently this layer is limited since it relies on DataInterpolations.jl which doesn't work with GPU arrays. This will be fixed in the future by extending support to different basis functions.
source
Boltz.Layers.TensorProductLayer Type
julia
TensorProductLayer(basis_fns, out_dim::Int; init_weight = randn32)
Constructs the Tensor Product Layer, which takes as input an array of n tensor product basis, [B1,B2,…,Bn] a data point x, computes
zi=Wi,:⊙[B1(x1)⊗B2(x2)⊗⋯⊗Bn(xn)]
where W is the layer's weight, and returns [z1,…,zout].
Arguments
  • basis_fns: Array of TensorProductBasis [B1(n1),…,Bk(nk)], where k corresponds to the dimension of the input.
  • out_dim: Dimension of the output.
Keyword Arguments
  • init_weight: Initializer for the weight matrix. Defaults to randn32.
Limited Backend Support
Support for backends apart from CPU and CUDA is limited and slow due to limited support for kron in the backend.
source
Boltz.Layers.ViPosEmbedding Type
julia
ViPosEmbedding(embedding_size, number_patches; init = randn32)
Positional embedding layer used by many vision transformer-like models.
source
Boltz.Layers.VisionTransformerEncoder Type
julia
VisionTransformerEncoder(in_planes, depth, number_heads; mlp_ratio = 4.0f0,
-    dropout = 0.0f0)
Transformer as used in the base ViT architecture (Dosovitskiy et al., 2020).
Arguments
  • in_planes: number of input channels
  • depth: number of attention blocks
  • number_heads: number of attention heads
Keyword Arguments
  • mlp_ratio: ratio of MLP layers to the number of input channels
  • dropout_rate: dropout rate
source
Boltz.Layers.ConvBatchNormActivation Method
julia
ConvBatchNormActivation(kernel_size::Dims, (in_filters, out_filters)::Pair{Int, Int},
+true
source
Boltz.Layers.SplineLayer Type
julia
SplineLayer(in_dims, grid_min, grid_max, grid_step, basis::Type{Basis};
+    train_grid::Union{Val, Bool}=Val(false), init_saved_points=nothing)
Constructs a spline layer with the given basis function.
Arguments
  • in_dims: input dimensions of the layer. This must be a tuple of integers, to construct a flat vector of saved_points pass in ().
  • grid_min: minimum value of the grid.
  • grid_max: maximum value of the grid.
  • grid_step: step size of the grid.
  • basis: basis function to use for the interpolation. Currently only the basis functions from DataInterpolations.jl are supported:
    1. ConstantInterpolation
    2. LinearInterpolation
    3. QuadraticInterpolation
    4. QuadraticSpline
    5. CubicSpline
Keyword Arguments
  • train_grid: whether to train the grid or not.
  • init_saved_points: values of the function at multiples of the time step. Initialized by default to a random vector sampled from the unit normal. Alternatively, can take a function with the signature init_saved_points(rng, in_dims, grid_min, grid_max, grid_step).
Warning
Currently this layer is limited since it relies on DataInterpolations.jl which doesn't work with GPU arrays. This will be fixed in the future by extending support to different basis functions.
source
Boltz.Layers.TensorProductLayer Type
julia
TensorProductLayer(basis_fns, out_dim::Int; init_weight = randn32)
Constructs the Tensor Product Layer, which takes as input an array of n tensor product basis, [B1,B2,…,Bn] a data point x, computes
zi=Wi,:⊙[B1(x1)⊗B2(x2)⊗⋯⊗Bn(xn)]
where W is the layer's weight, and returns [z1,…,zout].
Arguments
  • basis_fns: Array of TensorProductBasis [B1(n1),…,Bk(nk)], where k corresponds to the dimension of the input.
  • out_dim: Dimension of the output.
Keyword Arguments
  • init_weight: Initializer for the weight matrix. Defaults to randn32.
Limited Backend Support
Support for backends apart from CPU and CUDA is limited and slow due to limited support for kron in the backend.
source
Boltz.Layers.ViPosEmbedding Type
julia
ViPosEmbedding(embedding_size, number_patches; init = randn32)
Positional embedding layer used by many vision transformer-like models.
source
Boltz.Layers.VisionTransformerEncoder Type
julia
VisionTransformerEncoder(in_planes, depth, number_heads; mlp_ratio = 4.0f0,
+    dropout = 0.0f0)
Transformer as used in the base ViT architecture (Dosovitskiy et al., 2020).
Arguments
  • in_planes: number of input channels
  • depth: number of attention blocks
  • number_heads: number of attention heads
Keyword Arguments
  • mlp_ratio: ratio of MLP layers to the number of input channels
  • dropout_rate: dropout rate
source
Boltz.Layers.ConvBatchNormActivation Method
julia
ConvBatchNormActivation(kernel_size::Dims, (in_filters, out_filters)::Pair{Int, Int},
     depth::Int, act::F; use_norm::Bool=true, conv_kwargs=(;),
-    last_layer_activation::Bool=true, norm_kwargs=(;)) where {F}
This function is a convenience wrapper around ConvNormActivation that constructs a chain with norm_layer set to Lux.BatchNorm if use_norm is true and nothing otherwise. In most cases, users should use ConvNormActivation directly for a more flexible interface.
source
Bibliography ​
  • Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S. and others (2020). An image is worth 16x16 words: Transformers for image recognition at scale, arXiv preprint arXiv:2010.11929.
  • Greydanus, S.; Dzamba, M. and Yosinski, J. (2019). Hamiltonian neural networks. Advances in neural information processing systems 32.

-    
+    last_layer_activation::Bool=true, norm_kwargs=(;)) where {F}

This function is a convenience wrapper around ConvNormActivation that constructs a chain with norm_layer set to Lux.BatchNorm if use_norm is true and nothing otherwise. In most cases, users should use ConvNormActivation directly for a more flexible interface.

source

Bibliography ​

  • Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S. and others (2020). An image is worth 16x16 words: Transformers for image recognition at scale, arXiv preprint arXiv:2010.11929.

  • Greydanus, S.; Dzamba, M. and Yosinski, J. (2019). Hamiltonian neural networks. Advances in neural information processing systems 32.

+ \ No newline at end of file diff --git a/previews/PR98/api/private.html b/previews/PR98/api/private.html index 35620e4..1b4240e 100644 --- a/previews/PR98/api/private.html +++ b/previews/PR98/api/private.html @@ -9,11 +9,11 @@ - + - + - + @@ -25,8 +25,8 @@ -
Boltz.jl Docs

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Private API ​

This is the private API reference for Boltz.jl. You know what this means. Don't use these functions!

Boltz.Utils.fast_chunk Method
julia
fast_chunk(x::AbstractArray, ::Val{n}, ::Val{dim})

Type-stable and faster version of MLUtils.chunk.

source

Boltz.Utils.flatten_spatial Method
julia
flatten_spatial(x::AbstractArray{T, 4})

Flattens the first 2 dimensions of x, and permutes the remaining dimensions to (2, 1, 3).

source

Boltz.Utils.second_dim_mean Method
julia
second_dim_mean(x)

Computes the mean of x along dimension 2.

source

Boltz.Utils.should_type_assert Method
julia
should_type_assert(x)

In certain cases, to ensure type-stability we want to add type-asserts. But this won't work for exotic types like ForwardDiff.Dual. We use this function to check if we should add a type-assert for x.

source

- +
Boltz.jl Docs

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Private API ​

This is the private API reference for Boltz.jl. You know what this means. Don't use these functions!

Boltz.Utils.fast_chunk Method
julia
fast_chunk(x::AbstractArray, ::Val{n}, ::Val{dim})

Type-stable and faster version of MLUtils.chunk.

source

Boltz.Utils.flatten_spatial Method
julia
flatten_spatial(x::AbstractArray{T, 4})

Flattens the first 2 dimensions of x, and permutes the remaining dimensions to (2, 1, 3).

source

Boltz.Utils.second_dim_mean Method
julia
second_dim_mean(x)

Computes the mean of x along dimension 2.

source

Boltz.Utils.should_type_assert Method
julia
should_type_assert(x)

In certain cases, to ensure type-stability we want to add type-asserts. But this won't work for exotic types like ForwardDiff.Dual. We use this function to check if we should add a type-assert for x.

source

+ \ No newline at end of file diff --git a/previews/PR98/api/vision.html b/previews/PR98/api/vision.html index 7ee112b..2d9b497 100644 --- a/previews/PR98/api/vision.html +++ b/previews/PR98/api/vision.html @@ -9,11 +9,11 @@ - + - + - + @@ -25,8 +25,8 @@ -
Boltz.jl Docs

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Computer Vision Models (Vision API) ​

Native Lux Models ​

Boltz.Vision.AlexNet Type
julia
AlexNet(; kwargs...)

Create an AlexNet model (Krizhevsky et al., 2012).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.VGG Type
julia
VGG(imsize; config, inchannels, batchnorm = false, nclasses, fcsize, dropout)

Create a VGG model (Simonyan, 2014).

Arguments

  • imsize: input image width and height as a tuple

  • config: the configuration for the convolution layers

  • inchannels: number of input channels

  • batchnorm: set to true to use batch normalization after each convolution

  • nclasses: number of output classes

  • fcsize: intermediate fully connected layer size

  • dropout: dropout level between fully connected layers

source

julia
VGG(depth::Int; batchnorm::Bool=false, pretrained::Bool=false)

Create a VGG model (Simonyan, 2014) with ImageNet Configuration.

Arguments

  • depth::Int: the depth of the VGG model. Choices: {11, 13, 16, 19}.

Keyword Arguments

  • batchnorm = false: set to true to use batch normalization after each convolution.

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.VisionTransformer Type
julia
VisionTransformer(name::Symbol; pretrained=false)

Creates a Vision Transformer model with the specified configuration.

Arguments

  • name::Symbol: name of the Vision Transformer model to create. The following models are available – :tiny, :small, :base, :large, :huge, :giant, :gigantic.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Imported from Metalhead.jl ​

Load Metalhead

You need to load Metalhead before using these models.

Boltz.Vision.ConvMixer Function
julia
ConvMixer(name::Symbol; pretrained::Bool=false)

Create a ConvMixer model (Trockman and Kolter, 2022).

Arguments

  • name::Symbol: The name of the ConvMixer model. Must be one of :base, :small, or :large.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.DenseNet Function
julia
DenseNet(depth::Int; pretrained::Bool=false)

Create a DenseNet model (Huang et al., 2017).

Arguments

  • depth::Int: The depth of the DenseNet model. Must be one of 121, 161, 169, or 201.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.GoogLeNet Function
julia
GoogLeNet(; pretrained::Bool=false)

Create a GoogLeNet model (Szegedy et al., 2015).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.MobileNet Function
julia
MobileNet(name::Symbol; pretrained::Bool=false)

Create a MobileNet model (Howard, 2017; Sandler et al., 2018; Howard et al., 2019).

Arguments

  • name::Symbol: The name of the MobileNet model. Must be one of :v1, :v2, :v3_small, or :v3_large.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.ResNet Function
julia
ResNet(depth::Int; pretrained::Bool=false)

Create a ResNet model (He et al., 2016).

Arguments

  • depth::Int: The depth of the ResNet model. Must be one of 18, 34, 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.ResNeXt Function
julia
ResNeXt(depth::Int; cardinality=32, base_width=nothing, pretrained::Bool=false)

Create a ResNeXt model (Xie et al., 2017).

Arguments

  • depth::Int: The depth of the ResNeXt model. Must be one of 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

  • cardinality: The cardinality of the ResNeXt model. Defaults to 32.

  • base_width: The base width of the ResNeXt model. Defaults to 8 for depth 101 and 4 otherwise.

source

Boltz.Vision.SqueezeNet Function
julia
SqueezeNet(; pretrained::Bool=false)

Create a SqueezeNet model (Iandola et al., 2016).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.WideResNet Function
julia
WideResNet(depth::Int; pretrained::Bool=false)

Create a WideResNet model (Zagoruyko and Komodakis, 2017).

Arguments

  • depth::Int: The depth of the WideResNet model. Must be one of 18, 34, 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Pretrained Models ​

Load JLD2

You need to load JLD2 before being able to load pretrained weights.

Load Pretrained Weights

Pass pretrained=true to the model constructor to load the pretrained weights.

MODELTOP 1 ACCURACY (%)TOP 5 ACCURACY (%)
AlexNet()54.4877.72
VGG(11)67.3587.91
VGG(13)68.4088.48
VGG(16)70.2489.80
VGG(19)71.0990.27
VGG(11; batchnorm=true)69.0988.94
VGG(13; batchnorm=true)69.6689.49
VGG(16; batchnorm=true)72.1191.02
VGG(19; batchnorm=true)72.9591.32
ResNet(18)--
ResNet(34)--
ResNet(50)--
ResNet(101)--
ResNet(152)--
ResNeXt(50; cardinality=32, base_width=4)--
ResNeXt(101; cardinality=32, base_width=8)--
ResNeXt(101; cardinality=64, base_width=4)--
SqueezeNet()--
WideResNet(50)--
WideResNet(101)--

Pretrained Models from Metalhead

For Models imported from Metalhead, the pretrained weights can be loaded if they are available in Metalhead. Refer to the Metalhead.jl docs for a list of available pretrained models.

Preprocessing ​

All the pretrained models require that the images be normalized with the parameters mean = [0.485f0, 0.456f0, 0.406f0] and std = [0.229f0, 0.224f0, 0.225f0].

Bibliography ​

  • He, K.; Zhang, X.; Ren, S. and Sun, J. (2016). Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 770–778.

  • Howard, A.; Sandler, M.; Chu, G.; Chen, L.-C.; Chen, B.; Tan, M.; Wang, W.; Zhu, Y.; Pang, R.; Vasudevan, V. and others (2019). Searching for mobilenetv3. In: Proceedings of the IEEE/CVF international conference on computer vision; pp. 1314–1324.

  • Howard, A. G. (2017). MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications, arXiv preprint arXiv:1704.04861.

  • Huang, G.; Liu, Z.; Van Der Maaten, L. and Weinberger, K. Q. (2017). Densely connected convolutional networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 4700–4708.

  • Iandola, F. N.; Han, S.; Moskewicz, M. W.; Ashraf, K.; Dally, W. J. and Keutzer, K. (2016). SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size, arXiv:1602.07360 [cs.CV].

  • Krizhevsky, A.; Sutskever, I. and Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems 25.

  • Sandler, M.; Howard, A.; Zhu, M.; Zhmoginov, A. and Chen, L.-C. (2018). Mobilenetv2: Inverted residuals and linear bottlenecks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 4510–4520.

  • Simonyan, K. (2014). Very deep convolutional networks for large-scale image recognition, arXiv preprint arXiv:1409.1556.

  • Szegedy, C.; Liu, W.; Jia, Y.; Sermanet, P.; Reed, S.; Anguelov, D.; Erhan, D.; Vanhoucke, V. and Rabinovich, A. (2015). Going deeper with convolutions. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 1–9.

  • Trockman, A. and Kolter, J. Z. (2022). Patches are all you need? arXiv preprint arXiv:2201.09792.

  • Xie, S.; Girshick, R.; Dollár, P.; Tu, Z. and He, K. (2017). Aggregated residual transformations for deep neural networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 1492–1500.

  • Zagoruyko, S. and Komodakis, N. (2017). Wide Residual Networks, arXiv:1605.07146 [cs.CV].

- +
Boltz.jl Docs

Appearance


Trusted by

Scientific Computing

SciML.ai

Machine Learning

Computer Vision Models (Vision API) ​

Native Lux Models ​

Boltz.Vision.AlexNet Type
julia
AlexNet(; kwargs...)

Create an AlexNet model (Krizhevsky et al., 2012).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.VGG Type
julia
VGG(imsize; config, inchannels, batchnorm = false, nclasses, fcsize, dropout)

Create a VGG model (Simonyan, 2014).

Arguments

  • imsize: input image width and height as a tuple

  • config: the configuration for the convolution layers

  • inchannels: number of input channels

  • batchnorm: set to true to use batch normalization after each convolution

  • nclasses: number of output classes

  • fcsize: intermediate fully connected layer size

  • dropout: dropout level between fully connected layers

source

julia
VGG(depth::Int; batchnorm::Bool=false, pretrained::Bool=false)

Create a VGG model (Simonyan, 2014) with ImageNet Configuration.

Arguments

  • depth::Int: the depth of the VGG model. Choices: {11, 13, 16, 19}.

Keyword Arguments

  • batchnorm = false: set to true to use batch normalization after each convolution.

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.VisionTransformer Type
julia
VisionTransformer(name::Symbol; pretrained=false)

Creates a Vision Transformer model with the specified configuration.

Arguments

  • name::Symbol: name of the Vision Transformer model to create. The following models are available – :tiny, :small, :base, :large, :huge, :giant, :gigantic.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Imported from Metalhead.jl ​

Load Metalhead

You need to load Metalhead before using these models.

Boltz.Vision.ConvMixer Function
julia
ConvMixer(name::Symbol; pretrained::Bool=false)

Create a ConvMixer model (Trockman and Kolter, 2022).

Arguments

  • name::Symbol: The name of the ConvMixer model. Must be one of :base, :small, or :large.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.DenseNet Function
julia
DenseNet(depth::Int; pretrained::Bool=false)

Create a DenseNet model (Huang et al., 2017).

Arguments

  • depth::Int: The depth of the DenseNet model. Must be one of 121, 161, 169, or 201.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.GoogLeNet Function
julia
GoogLeNet(; pretrained::Bool=false)

Create a GoogLeNet model (Szegedy et al., 2015).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.MobileNet Function
julia
MobileNet(name::Symbol; pretrained::Bool=false)

Create a MobileNet model (Howard, 2017; Sandler et al., 2018; Howard et al., 2019).

Arguments

  • name::Symbol: The name of the MobileNet model. Must be one of :v1, :v2, :v3_small, or :v3_large.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.ResNet Function
julia
ResNet(depth::Int; pretrained::Bool=false)

Create a ResNet model (He et al., 2016).

Arguments

  • depth::Int: The depth of the ResNet model. Must be one of 18, 34, 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.ResNeXt Function
julia
ResNeXt(depth::Int; cardinality=32, base_width=nothing, pretrained::Bool=false)

Create a ResNeXt model (Xie et al., 2017).

Arguments

  • depth::Int: The depth of the ResNeXt model. Must be one of 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

  • cardinality: The cardinality of the ResNeXt model. Defaults to 32.

  • base_width: The base width of the ResNeXt model. Defaults to 8 for depth 101 and 4 otherwise.

source

Boltz.Vision.SqueezeNet Function
julia
SqueezeNet(; pretrained::Bool=false)

Create a SqueezeNet model (Iandola et al., 2016).

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Boltz.Vision.WideResNet Function
julia
WideResNet(depth::Int; pretrained::Bool=false)

Create a WideResNet model (Zagoruyko and Komodakis, 2017).

Arguments

  • depth::Int: The depth of the WideResNet model. Must be one of 18, 34, 50, 101, or 152.

Keyword Arguments

  • pretrained::Bool=false: If true, loads pretrained weights when LuxCore.setup is called.

source

Pretrained Models ​

Load JLD2

You need to load JLD2 before being able to load pretrained weights.

Load Pretrained Weights

Pass pretrained=true to the model constructor to load the pretrained weights.

MODELTOP 1 ACCURACY (%)TOP 5 ACCURACY (%)
AlexNet()54.4877.72
VGG(11)67.3587.91
VGG(13)68.4088.48
VGG(16)70.2489.80
VGG(19)71.0990.27
VGG(11; batchnorm=true)69.0988.94
VGG(13; batchnorm=true)69.6689.49
VGG(16; batchnorm=true)72.1191.02
VGG(19; batchnorm=true)72.9591.32
ResNet(18)--
ResNet(34)--
ResNet(50)--
ResNet(101)--
ResNet(152)--
ResNeXt(50; cardinality=32, base_width=4)--
ResNeXt(101; cardinality=32, base_width=8)--
ResNeXt(101; cardinality=64, base_width=4)--
SqueezeNet()--
WideResNet(50)--
WideResNet(101)--

Pretrained Models from Metalhead

For Models imported from Metalhead, the pretrained weights can be loaded if they are available in Metalhead. Refer to the Metalhead.jl docs for a list of available pretrained models.

Preprocessing ​

All the pretrained models require that the images be normalized with the parameters mean = [0.485f0, 0.456f0, 0.406f0] and std = [0.229f0, 0.224f0, 0.225f0].

Bibliography ​

  • He, K.; Zhang, X.; Ren, S. and Sun, J. (2016). Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 770–778.

  • Howard, A.; Sandler, M.; Chu, G.; Chen, L.-C.; Chen, B.; Tan, M.; Wang, W.; Zhu, Y.; Pang, R.; Vasudevan, V. and others (2019). Searching for mobilenetv3. In: Proceedings of the IEEE/CVF international conference on computer vision; pp. 1314–1324.

  • Howard, A. G. (2017). MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications, arXiv preprint arXiv:1704.04861.

  • Huang, G.; Liu, Z.; Van Der Maaten, L. and Weinberger, K. Q. (2017). Densely connected convolutional networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 4700–4708.

  • Iandola, F. N.; Han, S.; Moskewicz, M. W.; Ashraf, K.; Dally, W. J. and Keutzer, K. (2016). SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size, arXiv:1602.07360 [cs.CV].

  • Krizhevsky, A.; Sutskever, I. and Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems 25.

  • Sandler, M.; Howard, A.; Zhu, M.; Zhmoginov, A. and Chen, L.-C. (2018). Mobilenetv2: Inverted residuals and linear bottlenecks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 4510–4520.

  • Simonyan, K. (2014). Very deep convolutional networks for large-scale image recognition, arXiv preprint arXiv:1409.1556.

  • Szegedy, C.; Liu, W.; Jia, Y.; Sermanet, P.; Reed, S.; Anguelov, D.; Erhan, D.; Vanhoucke, V. and Rabinovich, A. (2015). Going deeper with convolutions. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 1–9.

  • Trockman, A. and Kolter, J. Z. (2022). Patches are all you need? arXiv preprint arXiv:2201.09792.

  • Xie, S.; Girshick, R.; Dollár, P.; Tu, Z. and He, K. (2017). Aggregated residual transformations for deep neural networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition; pp. 1492–1500.

  • Zagoruyko, S. and Komodakis, N. (2017). Wide Residual Networks, arXiv:1605.07146 [cs.CV].

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Chebyshev(n; dim::Int=1)
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julia
Cos(n; dim::Int=1)
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julia
Fourier(n; dim=1)

Constructs a Fourier basis of the form

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julia
Legendre(n; dim::Int=1)
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julia
ClassTokens(dim; init=zeros32)

Appends class tokens to an input with embedding dimension dim for use in many vision transformer models.

source

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julia
ConvNormActivation(kernel_size::Dims, in_chs::Integer, hidden_chs::Dims{N},
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julia
ClassTokens(dim; init=zeros32)
Appends class tokens to an input with embedding dimension dim for use in many vision transformer models.
source
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julia
ConvNormActivation(kernel_size::Dims, in_chs::Integer, hidden_chs::Dims{N},
     activation; norm_layer=nothing, conv_kwargs=(;), norm_kwargs=(;),
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Construct a Chain of convolutional layers with normalization and activation functions.
Arguments
  • kernel_size: size of the convolutional kernel
  • in_chs: number of input channels
  • hidden_chs: dimensions of the hidden layers
  • activation: activation function
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • conv_kwargs: keyword arguments for the convolutional layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
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julia
DynamicExpressionsLayer(operator_enum::OperatorEnum, expressions::Node...;
+    last_layer_activation::Bool=false) where {N}
Construct a Chain of convolutional layers with normalization and activation functions.
Arguments
  • kernel_size: size of the convolutional kernel
  • in_chs: number of input channels
  • hidden_chs: dimensions of the hidden layers
  • activation: activation function
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • conv_kwargs: keyword arguments for the convolutional layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
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julia
DynamicExpressionsLayer(operator_enum::OperatorEnum, expressions::Node...;
     eval_options::EvalOptions=EvalOptions())
 DynamicExpressionsLayer(operator_enum::OperatorEnum,
     expressions::AbstractVector{<:Node}; kwargs...)
Wraps a DynamicExpressions.jl Node into a Lux layer and allows the constant nodes to be updated using any of the AD Backends.
For details about these expressions, refer to the DynamicExpressions.jl documentation.
Arguments
  • operator_enum: OperatorEnum from DynamicExpressions.jl
  • expressions: Node from DynamicExpressions.jl or AbstractVector{<:Node}
Keyword Arguments
  • turbo: Use LoopVectorization.jl for faster evaluation (Deprecated)
  • bumper: Use Bumper.jl for faster evaluation (Deprecated)
  • eval_options: EvalOptions from DynamicExpressions.jl
These options are simply forwarded to DynamicExpressions.jl's eval_tree_array and eval_grad_tree_array function.
Extended Help
Example
julia
julia> operators = OperatorEnum(; binary_operators=[+, -, *], unary_operators=[cos]);
@@ -38,19 +38,19 @@ import{_ as p,c as l,j as s,a,G as n,a2 as t,B as r,o as h}from"./chunks/framewo
 true
 
 julia> ∂ps.layer_1.layer_2.params ≈ Float32[-31.0, 90.0]
-true
source
`,12))]),s("details",Q,[s("summary",null,[i[9]||(i[9]=s("a",{id:"Boltz.Layers.HamiltonianNN",href:"#Boltz.Layers.HamiltonianNN"},[s("span",{class:"jlbinding"},"Boltz.Layers.HamiltonianNN")],-1)),i[10]||(i[10]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[11]||(i[11]=t('
julia
HamiltonianNN{FST}(model; autodiff=nothing) where {FST}
Constructs a Hamiltonian Neural Network (Greydanus et al., 2019). This neural network is useful for learning symmetries and conservation laws by supervision on the gradients of the trajectories. It takes as input a concatenated vector of length 2n containing the position (of size n) and momentum (of size n) of the particles. It then returns the time derivatives for position and momentum.
Arguments
  • FST: If true, then the type of the state returned by the model must be same as the type of the input state. See the documentation on StatefulLuxLayer for more information.
  • model: A Lux.AbstractLuxLayer neural network that returns the Hamiltonian of the system. The model must return a "batched scalar", i.e. all the dimensions of the output except the last one must be equal to 1. The last dimension must be equal to the batchsize of the input.
Keyword Arguments
  • autodiff: The autodiff framework to be used for the internal Hamiltonian computation. The default is nothing, which selects the best possible backend available. The available options are AutoForwardDiff and AutoZygote.
Autodiff Backends
autodiffPackage NeededNotes
AutoZygoteZygote.jlPreferred Backend. Chosen if Zygote is loaded and autodiff is nothing.
AutoForwardDiffChosen if Zygote is not loaded and autodiff is nothing.
Note
This layer uses nested autodiff. Please refer to the manual entry on Nested Autodiff for more information and known limitations.
source
',10))]),s("details",g,[s("summary",null,[i[12]||(i[12]=s("a",{id:"Boltz.Layers.MLP",href:"#Boltz.Layers.MLP"},[s("span",{class:"jlbinding"},"Boltz.Layers.MLP")],-1)),i[13]||(i[13]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[14]||(i[14]=t(`
julia
MLP(in_dims::Integer, hidden_dims::Dims{N}, activation=NNlib.relu; norm_layer=nothing,
+true
source
`,12))]),s("details",Q,[s("summary",null,[i[9]||(i[9]=s("a",{id:"Boltz.Layers.HamiltonianNN",href:"#Boltz.Layers.HamiltonianNN"},[s("span",{class:"jlbinding"},"Boltz.Layers.HamiltonianNN")],-1)),i[10]||(i[10]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[11]||(i[11]=t('
julia
HamiltonianNN{FST}(model; autodiff=nothing) where {FST}
Constructs a Hamiltonian Neural Network (Greydanus et al., 2019). This neural network is useful for learning symmetries and conservation laws by supervision on the gradients of the trajectories. It takes as input a concatenated vector of length 2n containing the position (of size n) and momentum (of size n) of the particles. It then returns the time derivatives for position and momentum.
Arguments
  • FST: If true, then the type of the state returned by the model must be same as the type of the input state. See the documentation on StatefulLuxLayer for more information.
  • model: A Lux.AbstractLuxLayer neural network that returns the Hamiltonian of the system. The model must return a "batched scalar", i.e. all the dimensions of the output except the last one must be equal to 1. The last dimension must be equal to the batchsize of the input.
Keyword Arguments
  • autodiff: The autodiff framework to be used for the internal Hamiltonian computation. The default is nothing, which selects the best possible backend available. The available options are AutoForwardDiff and AutoZygote.
Autodiff Backends
autodiffPackage NeededNotes
AutoZygoteZygote.jlPreferred Backend. Chosen if Zygote is loaded and autodiff is nothing.
AutoForwardDiffChosen if Zygote is not loaded and autodiff is nothing.
Note
This layer uses nested autodiff. Please refer to the manual entry on Nested Autodiff for more information and known limitations.
source
',10))]),s("details",g,[s("summary",null,[i[12]||(i[12]=s("a",{id:"Boltz.Layers.MLP",href:"#Boltz.Layers.MLP"},[s("span",{class:"jlbinding"},"Boltz.Layers.MLP")],-1)),i[13]||(i[13]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[14]||(i[14]=t(`
julia
MLP(in_dims::Integer, hidden_dims::Dims{N}, activation=NNlib.relu; norm_layer=nothing,
     dropout_rate::Real=0.0f0, dense_kwargs=(;), norm_kwargs=(;),
-    last_layer_activation=false) where {N}
Construct a multi-layer perceptron (MLP) with dense layers, optional normalization layers, and dropout.
Arguments
  • in_dims: number of input dimensions
  • hidden_dims: dimensions of the hidden layers
  • activation: activation function (stacked after the normalization layer, if present else after the dense layer)
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • dropout_rate: dropout rate (default: 0.0f0)
  • dense_kwargs: keyword arguments for the dense layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
`,7))]),s("details",y,[s("summary",null,[i[15]||(i[15]=s("a",{id:"Boltz.Layers.MultiHeadSelfAttention",href:"#Boltz.Layers.MultiHeadSelfAttention"},[s("span",{class:"jlbinding"},"Boltz.Layers.MultiHeadSelfAttention")],-1)),i[16]||(i[16]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[17]||(i[17]=t(`
julia
MultiHeadSelfAttention(in_planes::Int, number_heads::Int; use_qkv_bias::Bool=false,
-    attention_dropout_rate::T=0.0f0, projection_dropout_rate::T=0.0f0)
Multi-head self-attention layer
Arguments
  • planes: number of input channels
  • nheads: number of heads
  • use_qkv_bias: whether to use bias in the layer to get the query, key and value
  • attn_dropout_prob: dropout probability after the self-attention layer
  • proj_dropout_prob: dropout probability after the projection layer
source
`,5))]),s("details",m,[s("summary",null,[i[18]||(i[18]=s("a",{id:"Boltz.Layers.PatchEmbedding",href:"#Boltz.Layers.PatchEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.PatchEmbedding")],-1)),i[19]||(i[19]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[20]||(i[20]=t(`
julia
PatchEmbedding(image_size, patch_size, in_channels, embed_planes;
-    norm_layer=Returns(Lux.NoOpLayer()), flatten=true)
Constructs a patch embedding layer with the given image size, patch size, input channels, and embedding planes. The patch size must be a divisor of the image size.
Arguments
  • image_size: image size as a tuple
  • patch_size: patch size as a tuple
  • in_channels: number of input channels
  • embed_planes: number of embedding planes
Keyword Arguments
  • norm_layer: Takes the embedding planes as input and returns a layer that normalizes the embedding planes. Defaults to no normalization.
  • flatten: set to true to flatten the output of the convolutional layer
source
`,7))]),s("details",c,[s("summary",null,[i[21]||(i[21]=s("a",{id:"Boltz.Layers.PeriodicEmbedding",href:"#Boltz.Layers.PeriodicEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.PeriodicEmbedding")],-1)),i[22]||(i[22]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[23]||(i[23]=t(`
julia
PeriodicEmbedding(idxs, periods)
Create an embedding periodic in some inputs with specified periods. Input indices not in idxs are passed through unchanged, but inputs in idxs are moved to the end of the output and replaced with their sines, followed by their cosines (scaled appropriately to have the specified periods). This smooth embedding preserves phase information and enforces periodicity.
For example, layer = PeriodicEmbedding([2, 3], [3.0, 1.0]) will create a layer periodic in the second input with period 3.0 and periodic in the third input with period 1.0. In this case, layer([a, b, c, d], st) == ([a, d, sinpi(2 / 3.0 * b), sinpi(2 / 1.0 * c), cospi(2 / 3.0 * b), cospi(2 / 1.0 * c)], st).
Arguments
  • idxs: Indices of the periodic inputs
  • periods: Periods of the periodic inputs, in the same order as in idxs
Inputs
  • x must be an AbstractArray with issubset(idxs, axes(x, 1))
  • st must be a NamedTuple where st.k = 2 ./ periods, but on the same device as x
Returns
  • AbstractArray of size (size(x, 1) + length(idxs), ...) where ... are the other dimensions of x.
  • st, unchanged
Example
julia
julia> layer = Layers.PeriodicEmbedding([2], [4.0])
+    last_layer_activation=false) where {N}
Construct a multi-layer perceptron (MLP) with dense layers, optional normalization layers, and dropout.
Arguments
  • in_dims: number of input dimensions
  • hidden_dims: dimensions of the hidden layers
  • activation: activation function (stacked after the normalization layer, if present else after the dense layer)
Keyword Arguments
  • norm_layer: Function with signature f(i::Integer, dims::Integer, act::F; kwargs...). i is the location of the layer in the model, dims is the channel dimension of the input, and act is the activation function. kwargs are forwarded from the norm_kwargs input, The function should return a normalization layer. Defaults to nothing, which means no normalization layer is used
  • dropout_rate: dropout rate (default: 0.0f0)
  • dense_kwargs: keyword arguments for the dense layers
  • norm_kwargs: keyword arguments for the normalization layers
  • last_layer_activation: set to true to apply the activation function to the last layer
source
`,7))]),s("details",y,[s("summary",null,[i[15]||(i[15]=s("a",{id:"Boltz.Layers.MultiHeadSelfAttention",href:"#Boltz.Layers.MultiHeadSelfAttention"},[s("span",{class:"jlbinding"},"Boltz.Layers.MultiHeadSelfAttention")],-1)),i[16]||(i[16]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[17]||(i[17]=t(`
julia
MultiHeadSelfAttention(in_planes::Int, number_heads::Int; use_qkv_bias::Bool=false,
+    attention_dropout_rate::T=0.0f0, projection_dropout_rate::T=0.0f0)
Multi-head self-attention layer
Arguments
  • planes: number of input channels
  • nheads: number of heads
  • use_qkv_bias: whether to use bias in the layer to get the query, key and value
  • attn_dropout_prob: dropout probability after the self-attention layer
  • proj_dropout_prob: dropout probability after the projection layer
source
`,5))]),s("details",m,[s("summary",null,[i[18]||(i[18]=s("a",{id:"Boltz.Layers.PatchEmbedding",href:"#Boltz.Layers.PatchEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.PatchEmbedding")],-1)),i[19]||(i[19]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[20]||(i[20]=t(`
julia
PatchEmbedding(image_size, patch_size, in_channels, embed_planes;
+    norm_layer=Returns(Lux.NoOpLayer()), flatten=true)
Constructs a patch embedding layer with the given image size, patch size, input channels, and embedding planes. The patch size must be a divisor of the image size.
Arguments
  • image_size: image size as a tuple
  • patch_size: patch size as a tuple
  • in_channels: number of input channels
  • embed_planes: number of embedding planes
Keyword Arguments
  • norm_layer: Takes the embedding planes as input and returns a layer that normalizes the embedding planes. Defaults to no normalization.
  • flatten: set to true to flatten the output of the convolutional layer
source
`,7))]),s("details",c,[s("summary",null,[i[21]||(i[21]=s("a",{id:"Boltz.Layers.PeriodicEmbedding",href:"#Boltz.Layers.PeriodicEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.PeriodicEmbedding")],-1)),i[22]||(i[22]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[23]||(i[23]=t(`
julia
PeriodicEmbedding(idxs, periods)
Create an embedding periodic in some inputs with specified periods. Input indices not in idxs are passed through unchanged, but inputs in idxs are moved to the end of the output and replaced with their sines, followed by their cosines (scaled appropriately to have the specified periods). This smooth embedding preserves phase information and enforces periodicity.
For example, layer = PeriodicEmbedding([2, 3], [3.0, 1.0]) will create a layer periodic in the second input with period 3.0 and periodic in the third input with period 1.0. In this case, layer([a, b, c, d], st) == ([a, d, sinpi(2 / 3.0 * b), sinpi(2 / 1.0 * c), cospi(2 / 3.0 * b), cospi(2 / 1.0 * c)], st).
Arguments
  • idxs: Indices of the periodic inputs
  • periods: Periods of the periodic inputs, in the same order as in idxs
Inputs
  • x must be an AbstractArray with issubset(idxs, axes(x, 1))
  • st must be a NamedTuple where st.k = 2 ./ periods, but on the same device as x
Returns
  • AbstractArray of size (size(x, 1) + length(idxs), ...) where ... are the other dimensions of x.
  • st, unchanged
Example
julia
julia> layer = Layers.PeriodicEmbedding([2], [4.0])
 PeriodicEmbedding([2], [4.0])
 
 julia> ps, st = Lux.setup(Random.default_rng(), layer);
 
 julia> all(layer([1.1, 2.2, 3.3], ps, st)[1] .==
            [1.1, 3.3, sinpi(2 / 4.0 * 2.2), cospi(2 / 4.0 * 2.2)])
-true
source
`,12))]),s("details",E,[s("summary",null,[i[24]||(i[24]=s("a",{id:"Boltz.Layers.SplineLayer",href:"#Boltz.Layers.SplineLayer"},[s("span",{class:"jlbinding"},"Boltz.Layers.SplineLayer")],-1)),i[25]||(i[25]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[26]||(i[26]=t(`
julia
SplineLayer(in_dims, grid_min, grid_max, grid_step, basis::Type{Basis};
-    train_grid::Union{Val, Bool}=Val(false), init_saved_points=nothing)
Constructs a spline layer with the given basis function.
Arguments
  • in_dims: input dimensions of the layer. This must be a tuple of integers, to construct a flat vector of saved_points pass in ().
  • grid_min: minimum value of the grid.
  • grid_max: maximum value of the grid.
  • grid_step: step size of the grid.
  • basis: basis function to use for the interpolation. Currently only the basis functions from DataInterpolations.jl are supported:
    1. ConstantInterpolation
    2. LinearInterpolation
    3. QuadraticInterpolation
    4. QuadraticSpline
    5. CubicSpline
Keyword Arguments
  • train_grid: whether to train the grid or not.
  • init_saved_points: values of the function at multiples of the time step. Initialized by default to a random vector sampled from the unit normal. Alternatively, can take a function with the signature init_saved_points(rng, in_dims, grid_min, grid_max, grid_step).
Warning
Currently this layer is limited since it relies on DataInterpolations.jl which doesn't work with GPU arrays. This will be fixed in the future by extending support to different basis functions.
source
`,8))]),s("details",u,[s("summary",null,[i[27]||(i[27]=s("a",{id:"Boltz.Layers.TensorProductLayer",href:"#Boltz.Layers.TensorProductLayer"},[s("span",{class:"jlbinding"},"Boltz.Layers.TensorProductLayer")],-1)),i[28]||(i[28]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[51]||(i[51]=t('
julia
TensorProductLayer(basis_fns, out_dim::Int; init_weight = randn32)
',1)),s("p",null,[i[31]||(i[31]=a("Constructs the Tensor Product Layer, which takes as input an array of n tensor product basis, ")),s("mjx-container",F,[(h(),l("svg",f,i[29]||(i[29]=[t('',1)]))),i[30]||(i[30]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",null,"B"),s("mn",null,"1")]),s("mo",null,","),s("msub",null,[s("mi",null,"B"),s("mn",null,"2")]),s("mo",null,","),s("mo",null,"…"),s("mo",null,","),s("msub",null,[s("mi",null,"B"),s("mi",null,"n")]),s("mo",{stretchy:"false"},"]")])],-1))]),i[32]||(i[32]=a(" a data point x, 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")),s("mjx-container",B,[(h(),l("svg",v,i[44]||(i[44]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 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Keyword Arguments
  • init_weight: Initializer for the weight matrix. Defaults to randn32.
Limited Backend Support
Support for backends apart from CPU and CUDA is limited and slow due to limited support for kron in the backend.
source
',4))]),s("details",H,[s("summary",null,[i[54]||(i[54]=s("a",{id:"Boltz.Layers.ViPosEmbedding",href:"#Boltz.Layers.ViPosEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.ViPosEmbedding")],-1)),i[55]||(i[55]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[56]||(i[56]=t('
julia
ViPosEmbedding(embedding_size, number_patches; init = randn32)
Positional embedding layer used by many vision transformer-like models.
source
',3))]),s("details",j,[s("summary",null,[i[57]||(i[57]=s("a",{id:"Boltz.Layers.VisionTransformerEncoder",href:"#Boltz.Layers.VisionTransformerEncoder"},[s("span",{class:"jlbinding"},"Boltz.Layers.VisionTransformerEncoder")],-1)),i[58]||(i[58]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[59]||(i[59]=t(`
julia
VisionTransformerEncoder(in_planes, depth, number_heads; mlp_ratio = 4.0f0,
-    dropout = 0.0f0)
Transformer as used in the base ViT architecture (Dosovitskiy et al., 2020).
Arguments
  • in_planes: number of input channels
  • depth: number of attention blocks
  • number_heads: number of attention heads
Keyword Arguments
  • mlp_ratio: ratio of MLP layers to the number of input channels
  • dropout_rate: dropout rate
source
`,7))]),s("details",M,[s("summary",null,[i[60]||(i[60]=s("a",{id:"Boltz.Layers.ConvBatchNormActivation-Union{Tuple{F}, Tuple{NTuple{N, Int64} where N, Pair{Int64, Int64}, Int64, F}} where F",href:"#Boltz.Layers.ConvBatchNormActivation-Union{Tuple{F}, Tuple{NTuple{N, Int64} where N, Pair{Int64, Int64}, Int64, F}} where F"},[s("span",{class:"jlbinding"},"Boltz.Layers.ConvBatchNormActivation")],-1)),i[61]||(i[61]=a()),n(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[62]||(i[62]=t(`
julia
ConvBatchNormActivation(kernel_size::Dims, (in_filters, out_filters)::Pair{Int, Int},
+true
source
`,12))]),s("details",E,[s("summary",null,[i[24]||(i[24]=s("a",{id:"Boltz.Layers.SplineLayer",href:"#Boltz.Layers.SplineLayer"},[s("span",{class:"jlbinding"},"Boltz.Layers.SplineLayer")],-1)),i[25]||(i[25]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[26]||(i[26]=t(`
julia
SplineLayer(in_dims, grid_min, grid_max, grid_step, basis::Type{Basis};
+    train_grid::Union{Val, Bool}=Val(false), init_saved_points=nothing)
Constructs a spline layer with the given basis function.
Arguments
  • in_dims: input dimensions of the layer. This must be a tuple of integers, to construct a flat vector of saved_points pass in ().
  • grid_min: minimum value of the grid.
  • grid_max: maximum value of the grid.
  • grid_step: step size of the grid.
  • basis: basis function to use for the interpolation. Currently only the basis functions from DataInterpolations.jl are supported:
    1. ConstantInterpolation
    2. LinearInterpolation
    3. QuadraticInterpolation
    4. QuadraticSpline
    5. CubicSpline
Keyword Arguments
  • train_grid: whether to train the grid or not.
  • init_saved_points: values of the function at multiples of the time step. Initialized by default to a random vector sampled from the unit normal. Alternatively, can take a function with the signature init_saved_points(rng, in_dims, grid_min, grid_max, grid_step).
Warning
Currently this layer is limited since it relies on DataInterpolations.jl which doesn't work with GPU arrays. This will be fixed in the future by extending support to different basis functions.
source
`,8))]),s("details",u,[s("summary",null,[i[27]||(i[27]=s("a",{id:"Boltz.Layers.TensorProductLayer",href:"#Boltz.Layers.TensorProductLayer"},[s("span",{class:"jlbinding"},"Boltz.Layers.TensorProductLayer")],-1)),i[28]||(i[28]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[51]||(i[51]=t('
julia
TensorProductLayer(basis_fns, out_dim::Int; init_weight = randn32)
',1)),s("p",null,[i[31]||(i[31]=a("Constructs the Tensor Product Layer, which takes as input an array of n tensor product basis, ")),s("mjx-container",F,[(h(),l("svg",f,i[29]||(i[29]=[t('',1)]))),i[30]||(i[30]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",null,"B"),s("mn",null,"1")]),s("mo",null,","),s("msub",null,[s("mi",null,"B"),s("mn",null,"2")]),s("mo",null,","),s("mo",null,"…"),s("mo",null,","),s("msub",null,[s("mi",null,"B"),s("mi",null,"n")]),s("mo",{stretchy:"false"},"]")])],-1))]),i[32]||(i[32]=a(" a data point x, 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")),s("mjx-container",_,[(h(),l("svg",L,i[35]||(i[35]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D44A",d:"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 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")),s("mjx-container",B,[(h(),l("svg",v,i[44]||(i[44]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 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Keyword Arguments
  • init_weight: Initializer for the weight matrix. Defaults to randn32.
Limited Backend Support
Support for backends apart from CPU and CUDA is limited and slow due to limited support for kron in the backend.
source
',4))]),s("details",H,[s("summary",null,[i[54]||(i[54]=s("a",{id:"Boltz.Layers.ViPosEmbedding",href:"#Boltz.Layers.ViPosEmbedding"},[s("span",{class:"jlbinding"},"Boltz.Layers.ViPosEmbedding")],-1)),i[55]||(i[55]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[56]||(i[56]=t('
julia
ViPosEmbedding(embedding_size, number_patches; init = randn32)
Positional embedding layer used by many vision transformer-like models.
source
',3))]),s("details",j,[s("summary",null,[i[57]||(i[57]=s("a",{id:"Boltz.Layers.VisionTransformerEncoder",href:"#Boltz.Layers.VisionTransformerEncoder"},[s("span",{class:"jlbinding"},"Boltz.Layers.VisionTransformerEncoder")],-1)),i[58]||(i[58]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[59]||(i[59]=t(`
julia
VisionTransformerEncoder(in_planes, depth, number_heads; mlp_ratio = 4.0f0,
+    dropout = 0.0f0)
Transformer as used in the base ViT architecture (Dosovitskiy et al., 2020).
Arguments
  • in_planes: number of input channels
  • depth: number of attention blocks
  • number_heads: number of attention heads
Keyword Arguments
  • mlp_ratio: ratio of MLP layers to the number of input channels
  • dropout_rate: dropout rate
source
`,7))]),s("details",M,[s("summary",null,[i[60]||(i[60]=s("a",{id:"Boltz.Layers.ConvBatchNormActivation-Union{Tuple{F}, Tuple{NTuple{N, Int64} where N, Pair{Int64, Int64}, Int64, F}} where F",href:"#Boltz.Layers.ConvBatchNormActivation-Union{Tuple{F}, Tuple{NTuple{N, Int64} where N, Pair{Int64, Int64}, Int64, F}} where F"},[s("span",{class:"jlbinding"},"Boltz.Layers.ConvBatchNormActivation")],-1)),i[61]||(i[61]=a()),n(e,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[62]||(i[62]=t(`
julia
ConvBatchNormActivation(kernel_size::Dims, (in_filters, out_filters)::Pair{Int, Int},
     depth::Int, act::F; use_norm::Bool=true, conv_kwargs=(;),
-    last_layer_activation::Bool=true, norm_kwargs=(;)) where {F}
This function is a convenience wrapper around ConvNormActivation that constructs a chain with norm_layer set to Lux.BatchNorm if use_norm is true and nothing otherwise. In most cases, users should use ConvNormActivation directly for a more flexible interface.
source
`,3))]),i[65]||(i[65]=t('
Bibliography ​
  • Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S. and others (2020). An image is worth 16x16 words: Transformers for image recognition at scale, arXiv preprint arXiv:2010.11929.
  • Greydanus, S.; Dzamba, M. and Yosinski, J. (2019). Hamiltonian neural networks. Advances in neural information processing systems 32.
',3))])}const O=p(o,[["render",V]]);export{R as __pageData,O as default};
+    last_layer_activation::Bool=true, norm_kwargs=(;)) where {F}

This function is a convenience wrapper around ConvNormActivation that constructs a chain with norm_layer set to Lux.BatchNorm if use_norm is true and nothing otherwise. In most cases, users should use ConvNormActivation directly for a more flexible interface.

source

`,3))]),i[65]||(i[65]=t('

Bibliography ​

',3))])}const O=p(o,[["render",V]]);export{R as __pageData,O as default}; diff --git a/previews/PR98/assets/api_layers.md.DmVxh5An.lean.js b/previews/PR98/assets/api_layers.md.BdKhvUXl.lean.js similarity index 100% rename from previews/PR98/assets/api_layers.md.DmVxh5An.lean.js rename to previews/PR98/assets/api_layers.md.BdKhvUXl.lean.js diff --git a/previews/PR98/assets/api_private.md.BJWD8BVj.js b/previews/PR98/assets/api_private.md.D3-ym3uf.js similarity index 88% rename from previews/PR98/assets/api_private.md.BJWD8BVj.js rename to previews/PR98/assets/api_private.md.D3-ym3uf.js index 947b91a..6db4a1d 100644 --- a/previews/PR98/assets/api_private.md.BJWD8BVj.js +++ b/previews/PR98/assets/api_private.md.D3-ym3uf.js @@ -1 +1 @@ -import{_ as n,c as o,j as t,a as e,G as i,a2 as l,B as p,o as r}from"./chunks/framework.BI0fNMXE.js";const j=JSON.parse('{"title":"Private API","description":"","frontmatter":{},"headers":[],"relativePath":"api/private.md","filePath":"api/private.md","lastUpdated":null}'),d={name:"api/private.md"},h={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function b(_,s,g,f,y,T){const a=p("Badge");return r(),o("div",null,[s[12]||(s[12]=t("h1",{id:"Private-API",tabindex:"-1"},[e("Private API "),t("a",{class:"header-anchor",href:"#Private-API","aria-label":'Permalink to "Private API {#Private-API}"'},"​")],-1)),s[13]||(s[13]=t("p",null,"This is the private API reference for Boltz.jl. You know what this means. Don't use these functions!",-1)),t("details",h,[t("summary",null,[s[0]||(s[0]=t("a",{id:"Boltz.Utils.fast_chunk-Tuple{Int64, Int64}",href:"#Boltz.Utils.fast_chunk-Tuple{Int64, Int64}"},[t("span",{class:"jlbinding"},"Boltz.Utils.fast_chunk")],-1)),s[1]||(s[1]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=l('
julia
fast_chunk(x::AbstractArray, ::Val{n}, ::Val{dim})

Type-stable and faster version of MLUtils.chunk.

source

',3))]),t("details",u,[t("summary",null,[s[3]||(s[3]=t("a",{id:"Boltz.Utils.flatten_spatial-Union{Tuple{AbstractArray{T, 4}}, Tuple{T}} where T",href:"#Boltz.Utils.flatten_spatial-Union{Tuple{AbstractArray{T, 4}}, Tuple{T}} where T"},[t("span",{class:"jlbinding"},"Boltz.Utils.flatten_spatial")],-1)),s[4]||(s[4]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=l('
julia
flatten_spatial(x::AbstractArray{T, 4})

Flattens the first 2 dimensions of x, and permutes the remaining dimensions to (2, 1, 3).

source

',3))]),t("details",k,[t("summary",null,[s[6]||(s[6]=t("a",{id:"Boltz.Utils.second_dim_mean-Tuple{Any}",href:"#Boltz.Utils.second_dim_mean-Tuple{Any}"},[t("span",{class:"jlbinding"},"Boltz.Utils.second_dim_mean")],-1)),s[7]||(s[7]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=l('
julia
second_dim_mean(x)

Computes the mean of x along dimension 2.

source

',3))]),t("details",c,[t("summary",null,[s[9]||(s[9]=t("a",{id:"Boltz.Utils.should_type_assert-Union{Tuple{AbstractArray{T}}, Tuple{T}} where T",href:"#Boltz.Utils.should_type_assert-Union{Tuple{AbstractArray{T}}, Tuple{T}} where T"},[t("span",{class:"jlbinding"},"Boltz.Utils.should_type_assert")],-1)),s[10]||(s[10]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[11]||(s[11]=l('
julia
should_type_assert(x)

In certain cases, to ensure type-stability we want to add type-asserts. But this won't work for exotic types like ForwardDiff.Dual. We use this function to check if we should add a type-assert for x.

source

',3))])])}const A=n(d,[["render",b]]);export{j as __pageData,A as default}; +import{_ as n,c as o,j as t,a as e,G as i,a2 as l,B as p,o as r}from"./chunks/framework.BI0fNMXE.js";const j=JSON.parse('{"title":"Private API","description":"","frontmatter":{},"headers":[],"relativePath":"api/private.md","filePath":"api/private.md","lastUpdated":null}'),d={name:"api/private.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""};function _(b,s,g,f,y,T){const a=p("Badge");return r(),o("div",null,[s[12]||(s[12]=t("h1",{id:"Private-API",tabindex:"-1"},[e("Private API "),t("a",{class:"header-anchor",href:"#Private-API","aria-label":'Permalink to "Private API {#Private-API}"'},"​")],-1)),s[13]||(s[13]=t("p",null,"This is the private API reference for Boltz.jl. You know what this means. Don't use these functions!",-1)),t("details",h,[t("summary",null,[s[0]||(s[0]=t("a",{id:"Boltz.Utils.fast_chunk-Tuple{Int64, Int64}",href:"#Boltz.Utils.fast_chunk-Tuple{Int64, Int64}"},[t("span",{class:"jlbinding"},"Boltz.Utils.fast_chunk")],-1)),s[1]||(s[1]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=l('
julia
fast_chunk(x::AbstractArray, ::Val{n}, ::Val{dim})

Type-stable and faster version of MLUtils.chunk.

source

',3))]),t("details",c,[t("summary",null,[s[3]||(s[3]=t("a",{id:"Boltz.Utils.flatten_spatial-Union{Tuple{AbstractArray{T, 4}}, Tuple{T}} where T",href:"#Boltz.Utils.flatten_spatial-Union{Tuple{AbstractArray{T, 4}}, Tuple{T}} where T"},[t("span",{class:"jlbinding"},"Boltz.Utils.flatten_spatial")],-1)),s[4]||(s[4]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=l('
julia
flatten_spatial(x::AbstractArray{T, 4})

Flattens the first 2 dimensions of x, and permutes the remaining dimensions to (2, 1, 3).

source

',3))]),t("details",u,[t("summary",null,[s[6]||(s[6]=t("a",{id:"Boltz.Utils.second_dim_mean-Tuple{Any}",href:"#Boltz.Utils.second_dim_mean-Tuple{Any}"},[t("span",{class:"jlbinding"},"Boltz.Utils.second_dim_mean")],-1)),s[7]||(s[7]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=l('
julia
second_dim_mean(x)

Computes the mean of x along dimension 2.

source

',3))]),t("details",k,[t("summary",null,[s[9]||(s[9]=t("a",{id:"Boltz.Utils.should_type_assert-Union{Tuple{AbstractArray{T}}, Tuple{T}} where T",href:"#Boltz.Utils.should_type_assert-Union{Tuple{AbstractArray{T}}, Tuple{T}} where T"},[t("span",{class:"jlbinding"},"Boltz.Utils.should_type_assert")],-1)),s[10]||(s[10]=e()),i(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[11]||(s[11]=l('
julia
should_type_assert(x)

In certain cases, to ensure type-stability we want to add type-asserts. But this won't work for exotic types like ForwardDiff.Dual. We use this function to check if we should add a type-assert for x.

source

',3))])])}const A=n(d,[["render",_]]);export{j as __pageData,A as default}; diff --git a/previews/PR98/assets/api_private.md.BJWD8BVj.lean.js b/previews/PR98/assets/api_private.md.D3-ym3uf.lean.js similarity index 81% rename from previews/PR98/assets/api_private.md.BJWD8BVj.lean.js rename to previews/PR98/assets/api_private.md.D3-ym3uf.lean.js index 76c0fd4..dc365b8 100644 --- a/previews/PR98/assets/api_private.md.BJWD8BVj.lean.js +++ b/previews/PR98/assets/api_private.md.D3-ym3uf.lean.js @@ -1 +1 @@ -import{_ as n,c as o,j as t,a as e,G as i,a2 as l,B as p,o as r}from"./chunks/framework.BI0fNMXE.js";const j=JSON.parse('{"title":"Private API","description":"","frontmatter":{},"headers":[],"relativePath":"api/private.md","filePath":"api/private.md","lastUpdated":null}'),d={name:"api/private.md"},h={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function b(_,s,g,f,y,T){const a=p("Badge");return r(),o("div",null,[s[12]||(s[12]=t("h1",{id:"Private-API",tabindex:"-1"},[e("Private API "),t("a",{class:"header-anchor",href:"#Private-API","aria-label":'Permalink to "Private API {#Private-API}"'},"​")],-1)),s[13]||(s[13]=t("p",null,"This is the private API reference for Boltz.jl. 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julia
AlexNet(; kwargs...)

Create an AlexNet model (Krizhevsky et al., 2012).

Keyword Arguments

source

',5))]),t("details",c,[t("summary",null,[e[3]||(e[3]=t("a",{id:"Boltz.Vision.VGG",href:"#Boltz.Vision.VGG"},[t("span",{class:"jlbinding"},"Boltz.Vision.VGG")],-1)),e[4]||(e[4]=i()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[5]||(e[5]=s('
julia
VGG(imsize; config, inchannels, batchnorm = false, nclasses, fcsize, dropout)

Create a VGG model (Simonyan, 2014).

Arguments

source

julia
VGG(depth::Int; batchnorm::Bool=false, pretrained::Bool=false)

Create a VGG model (Simonyan, 2014) with ImageNet Configuration.

Arguments

Keyword Arguments

source

',12))]),t("details",g,[t("summary",null,[e[6]||(e[6]=t("a",{id:"Boltz.Vision.VisionTransformer",href:"#Boltz.Vision.VisionTransformer"},[t("span",{class:"jlbinding"},"Boltz.Vision.VisionTransformer")],-1)),e[7]||(e[7]=i()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[8]||(e[8]=s('
julia
VisionTransformer(name::Symbol; pretrained=false)

Creates a Vision Transformer model with the specified configuration.

Arguments

Keyword Arguments

source

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julia
ConvMixer(name::Symbol; pretrained::Bool=false)

Create a ConvMixer model (Trockman and Kolter, 2022).

Arguments

Keyword Arguments

source

',7))]),t("details",u,[t("summary",null,[e[12]||(e[12]=t("a",{id:"Boltz.Vision.DenseNet",href:"#Boltz.Vision.DenseNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.DenseNet")],-1)),e[13]||(e[13]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s('
julia
DenseNet(depth::Int; pretrained::Bool=false)

Create a DenseNet model (Huang et al., 2017).

Arguments

Keyword Arguments

source

',7))]),t("details",y,[t("summary",null,[e[15]||(e[15]=t("a",{id:"Boltz.Vision.GoogLeNet",href:"#Boltz.Vision.GoogLeNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.GoogLeNet")],-1)),e[16]||(e[16]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[17]||(e[17]=s('
julia
GoogLeNet(; pretrained::Bool=false)

Create a GoogLeNet model (Szegedy et al., 2015).

Keyword Arguments

source

',5))]),t("details",m,[t("summary",null,[e[18]||(e[18]=t("a",{id:"Boltz.Vision.MobileNet",href:"#Boltz.Vision.MobileNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.MobileNet")],-1)),e[19]||(e[19]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[20]||(e[20]=s('
julia
MobileNet(name::Symbol; pretrained::Bool=false)

Create a MobileNet model (Howard, 2017; Sandler et al., 2018; Howard et al., 2019).

Arguments

Keyword Arguments

source

',7))]),t("details",b,[t("summary",null,[e[21]||(e[21]=t("a",{id:"Boltz.Vision.ResNet",href:"#Boltz.Vision.ResNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.ResNet")],-1)),e[22]||(e[22]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[23]||(e[23]=s('
julia
ResNet(depth::Int; pretrained::Bool=false)

Create a ResNet model (He et al., 2016).

Arguments

Keyword Arguments

source

',7))]),t("details",f,[t("summary",null,[e[24]||(e[24]=t("a",{id:"Boltz.Vision.ResNeXt",href:"#Boltz.Vision.ResNeXt"},[t("span",{class:"jlbinding"},"Boltz.Vision.ResNeXt")],-1)),e[25]||(e[25]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[26]||(e[26]=s('
julia
ResNeXt(depth::Int; cardinality=32, base_width=nothing, pretrained::Bool=false)

Create a ResNeXt model (Xie et al., 2017).

Arguments

Keyword Arguments

source

',7))]),t("details",C,[t("summary",null,[e[27]||(e[27]=t("a",{id:"Boltz.Vision.SqueezeNet",href:"#Boltz.Vision.SqueezeNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.SqueezeNet")],-1)),e[28]||(e[28]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[29]||(e[29]=s('
julia
SqueezeNet(; pretrained::Bool=false)

Create a SqueezeNet model (Iandola et al., 2016).

Keyword Arguments

source

',5))]),t("details",v,[t("summary",null,[e[30]||(e[30]=t("a",{id:"Boltz.Vision.WideResNet",href:"#Boltz.Vision.WideResNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.WideResNet")],-1)),e[31]||(e[31]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[32]||(e[32]=s('
julia
WideResNet(depth::Int; pretrained::Bool=false)

Create a WideResNet model (Zagoruyko and Komodakis, 2017).

Arguments

Keyword Arguments

source

',7))]),e[37]||(e[37]=s('

Pretrained Models ​

Load JLD2

You need to load JLD2 before being able to load pretrained weights.

Load Pretrained Weights

Pass pretrained=true to the model constructor to load the pretrained weights.

MODELTOP 1 ACCURACY (%)TOP 5 ACCURACY (%)
AlexNet()54.4877.72
VGG(11)67.3587.91
VGG(13)68.4088.48
VGG(16)70.2489.80
VGG(19)71.0990.27
VGG(11; batchnorm=true)69.0988.94
VGG(13; batchnorm=true)69.6689.49
VGG(16; batchnorm=true)72.1191.02
VGG(19; batchnorm=true)72.9591.32
ResNet(18)--
ResNet(34)--
ResNet(50)--
ResNet(101)--
ResNet(152)--
ResNeXt(50; cardinality=32, base_width=4)--
ResNeXt(101; cardinality=32, base_width=8)--
ResNeXt(101; cardinality=64, base_width=4)--
SqueezeNet()--
WideResNet(50)--
WideResNet(101)--

Pretrained Models from Metalhead

For Models imported from Metalhead, the pretrained weights can be loaded if they are available in Metalhead. Refer to the Metalhead.jl docs for a list of available pretrained models.

Preprocessing ​

All the pretrained models require that the images be normalized with the parameters mean = [0.485f0, 0.456f0, 0.406f0] and std = [0.229f0, 0.224f0, 0.225f0].

Bibliography ​

',10))])}const V=n(p,[["render",E]]);export{T as __pageData,V as default}; +import{_ as n,c as o,j as t,a as i,G as l,a2 as s,B as r,o as d}from"./chunks/framework.BI0fNMXE.js";const T=JSON.parse('{"title":"Computer Vision Models (Vision API)","description":"","frontmatter":{},"headers":[],"relativePath":"api/vision.md","filePath":"api/vision.md","lastUpdated":null}'),p={name:"api/vision.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""};function E(x,e,F,B,_,j){const a=r("Badge");return d(),o("div",null,[e[33]||(e[33]=t("h1",{id:"Computer-Vision-Models-(Vision-API)",tabindex:"-1"},[i("Computer Vision Models ("),t("code",null,"Vision"),i(" API) "),t("a",{class:"header-anchor",href:"#Computer-Vision-Models-(Vision-API)","aria-label":'Permalink to "Computer Vision Models (`Vision` API) {#Computer-Vision-Models-(Vision-API)}"'},"​")],-1)),e[34]||(e[34]=t("h2",{id:"Native-Lux-Models",tabindex:"-1"},[i("Native Lux Models "),t("a",{class:"header-anchor",href:"#Native-Lux-Models","aria-label":'Permalink to "Native Lux Models {#Native-Lux-Models}"'},"​")],-1)),t("details",h,[t("summary",null,[e[0]||(e[0]=t("a",{id:"Boltz.Vision.AlexNet",href:"#Boltz.Vision.AlexNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.AlexNet")],-1)),e[1]||(e[1]=i()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[2]||(e[2]=s('
julia
AlexNet(; kwargs...)

Create an AlexNet model (Krizhevsky et al., 2012).

Keyword Arguments

source

',5))]),t("details",c,[t("summary",null,[e[3]||(e[3]=t("a",{id:"Boltz.Vision.VGG",href:"#Boltz.Vision.VGG"},[t("span",{class:"jlbinding"},"Boltz.Vision.VGG")],-1)),e[4]||(e[4]=i()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[5]||(e[5]=s('
julia
VGG(imsize; config, inchannels, batchnorm = false, nclasses, fcsize, dropout)

Create a VGG model (Simonyan, 2014).

Arguments

source

julia
VGG(depth::Int; batchnorm::Bool=false, pretrained::Bool=false)

Create a VGG model (Simonyan, 2014) with ImageNet Configuration.

Arguments

Keyword Arguments

source

',12))]),t("details",g,[t("summary",null,[e[6]||(e[6]=t("a",{id:"Boltz.Vision.VisionTransformer",href:"#Boltz.Vision.VisionTransformer"},[t("span",{class:"jlbinding"},"Boltz.Vision.VisionTransformer")],-1)),e[7]||(e[7]=i()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[8]||(e[8]=s('
julia
VisionTransformer(name::Symbol; pretrained=false)

Creates a Vision Transformer model with the specified configuration.

Arguments

Keyword Arguments

source

',7))]),e[35]||(e[35]=t("h2",{id:"Imported-from-Metalhead.jl",tabindex:"-1"},[i("Imported from Metalhead.jl "),t("a",{class:"header-anchor",href:"#Imported-from-Metalhead.jl","aria-label":'Permalink to "Imported from Metalhead.jl {#Imported-from-Metalhead.jl}"'},"​")],-1)),e[36]||(e[36]=t("div",{class:"tip custom-block"},[t("p",{class:"custom-block-title"},"Load Metalhead"),t("p",null,[i("You need to load "),t("code",null,"Metalhead"),i(" before using these models.")])],-1)),t("details",k,[t("summary",null,[e[9]||(e[9]=t("a",{id:"Boltz.Vision.ConvMixer",href:"#Boltz.Vision.ConvMixer"},[t("span",{class:"jlbinding"},"Boltz.Vision.ConvMixer")],-1)),e[10]||(e[10]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[11]||(e[11]=s('
julia
ConvMixer(name::Symbol; pretrained::Bool=false)

Create a ConvMixer model (Trockman and Kolter, 2022).

Arguments

Keyword Arguments

source

',7))]),t("details",u,[t("summary",null,[e[12]||(e[12]=t("a",{id:"Boltz.Vision.DenseNet",href:"#Boltz.Vision.DenseNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.DenseNet")],-1)),e[13]||(e[13]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[14]||(e[14]=s('
julia
DenseNet(depth::Int; pretrained::Bool=false)

Create a DenseNet model (Huang et al., 2017).

Arguments

Keyword Arguments

source

',7))]),t("details",y,[t("summary",null,[e[15]||(e[15]=t("a",{id:"Boltz.Vision.GoogLeNet",href:"#Boltz.Vision.GoogLeNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.GoogLeNet")],-1)),e[16]||(e[16]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[17]||(e[17]=s('
julia
GoogLeNet(; pretrained::Bool=false)

Create a GoogLeNet model (Szegedy et al., 2015).

Keyword Arguments

source

',5))]),t("details",m,[t("summary",null,[e[18]||(e[18]=t("a",{id:"Boltz.Vision.MobileNet",href:"#Boltz.Vision.MobileNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.MobileNet")],-1)),e[19]||(e[19]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[20]||(e[20]=s('
julia
MobileNet(name::Symbol; pretrained::Bool=false)

Create a MobileNet model (Howard, 2017; Sandler et al., 2018; Howard et al., 2019).

Arguments

Keyword Arguments

source

',7))]),t("details",b,[t("summary",null,[e[21]||(e[21]=t("a",{id:"Boltz.Vision.ResNet",href:"#Boltz.Vision.ResNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.ResNet")],-1)),e[22]||(e[22]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[23]||(e[23]=s('
julia
ResNet(depth::Int; pretrained::Bool=false)

Create a ResNet model (He et al., 2016).

Arguments

Keyword Arguments

source

',7))]),t("details",f,[t("summary",null,[e[24]||(e[24]=t("a",{id:"Boltz.Vision.ResNeXt",href:"#Boltz.Vision.ResNeXt"},[t("span",{class:"jlbinding"},"Boltz.Vision.ResNeXt")],-1)),e[25]||(e[25]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[26]||(e[26]=s('
julia
ResNeXt(depth::Int; cardinality=32, base_width=nothing, pretrained::Bool=false)

Create a ResNeXt model (Xie et al., 2017).

Arguments

Keyword Arguments

source

',7))]),t("details",C,[t("summary",null,[e[27]||(e[27]=t("a",{id:"Boltz.Vision.SqueezeNet",href:"#Boltz.Vision.SqueezeNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.SqueezeNet")],-1)),e[28]||(e[28]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[29]||(e[29]=s('
julia
SqueezeNet(; pretrained::Bool=false)

Create a SqueezeNet model (Iandola et al., 2016).

Keyword Arguments

source

',5))]),t("details",v,[t("summary",null,[e[30]||(e[30]=t("a",{id:"Boltz.Vision.WideResNet",href:"#Boltz.Vision.WideResNet"},[t("span",{class:"jlbinding"},"Boltz.Vision.WideResNet")],-1)),e[31]||(e[31]=i()),l(a,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[32]||(e[32]=s('
julia
WideResNet(depth::Int; pretrained::Bool=false)

Create a WideResNet model (Zagoruyko and Komodakis, 2017).

Arguments

Keyword Arguments

source

',7))]),e[37]||(e[37]=s('

Pretrained Models ​

Load JLD2

You need to load JLD2 before being able to load pretrained weights.

Load Pretrained Weights

Pass pretrained=true to the model constructor to load the pretrained weights.

MODELTOP 1 ACCURACY (%)TOP 5 ACCURACY (%)
AlexNet()54.4877.72
VGG(11)67.3587.91
VGG(13)68.4088.48
VGG(16)70.2489.80
VGG(19)71.0990.27
VGG(11; batchnorm=true)69.0988.94
VGG(13; batchnorm=true)69.6689.49
VGG(16; batchnorm=true)72.1191.02
VGG(19; batchnorm=true)72.9591.32
ResNet(18)--
ResNet(34)--
ResNet(50)--
ResNet(101)--
ResNet(152)--
ResNeXt(50; cardinality=32, base_width=4)--
ResNeXt(101; cardinality=32, base_width=8)--
ResNeXt(101; cardinality=64, base_width=4)--
SqueezeNet()--
WideResNet(50)--
WideResNet(101)--

Pretrained Models from Metalhead

For Models imported from Metalhead, the pretrained weights can be loaded if they are available in Metalhead. Refer to the Metalhead.jl docs for a list of available pretrained models.

Preprocessing ​

All the pretrained models require that the images be normalized with the parameters mean = [0.485f0, 0.456f0, 0.406f0] and std = [0.229f0, 0.224f0, 0.225f0].

Bibliography ​

',10))])}const V=n(p,[["render",E]]);export{T as __pageData,V as default}; diff --git a/previews/PR98/assets/api_vision.md.B5rI8T8X.lean.js b/previews/PR98/assets/api_vision.md.DUpvbNMg.lean.js similarity index 97% rename from previews/PR98/assets/api_vision.md.B5rI8T8X.lean.js rename to previews/PR98/assets/api_vision.md.DUpvbNMg.lean.js index 63b7383..c0c252f 100644 --- a/previews/PR98/assets/api_vision.md.B5rI8T8X.lean.js +++ b/previews/PR98/assets/api_vision.md.DUpvbNMg.lean.js @@ -1 +1 @@ -import{_ as n,c as o,j as t,a as i,G as l,a2 as s,B as d,o as r}from"./chunks/framework.BI0fNMXE.js";const T=JSON.parse('{"title":"Computer Vision Models (Vision API)","description":"","frontmatter":{},"headers":[],"relativePath":"api/vision.md","filePath":"api/vision.md","lastUpdated":null}'),p={name:"api/vision.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring 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__pageData,V as default}; +import{_ as n,c as o,j as t,a as i,G as l,a2 as s,B as r,o as d}from"./chunks/framework.BI0fNMXE.js";const T=JSON.parse('{"title":"Computer Vision Models (Vision API)","description":"","frontmatter":{},"headers":[],"relativePath":"api/vision.md","filePath":"api/vision.md","lastUpdated":null}'),p={name:"api/vision.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},v={class:"jldocstring custom-block",open:""};function E(x,e,F,B,_,j){const a=r("Badge");return d(),o("div",null,[e[33]||(e[33]=t("h1",{id:"Computer-Vision-Models-(Vision-API)",tabindex:"-1"},[i("Computer 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How to Install Boltz.jl? ​

Its easy to install Boltz.jl. Since Boltz.jl is registered in the Julia General registry, you can simply run the following command in the Julia REPL:

julia
julia> using Pkg
 julia> Pkg.add("Boltz")

If you want to use the latest unreleased version of Boltz.jl, you can run the following command: (in most cases the released version will be same as the version on github)

julia
julia> using Pkg
-julia> Pkg.add(url="https://github.com/LuxDL/Boltz.jl")

Want GPU Support? ​

Install the following package(s):

julia
using Pkg
+julia> Pkg.add(url="https://github.com/LuxDL/Boltz.jl")

Want GPU Support? ​

Install the following package(s):

julia
using Pkg
 Pkg.add("LuxCUDA")
 # or
 Pkg.add(["CUDA", "cuDNN"])
julia
using Pkg
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diff --git a/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.BLCy63T9.js b/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.js
similarity index 51%
rename from previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.BLCy63T9.js
rename to previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.js
index f5be45a..1403b39 100644
--- a/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.BLCy63T9.js
+++ b/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.js
@@ -1,561 +1,167 @@
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Solving Optimal Control Problems with Symbolic Universal Differential Equations ​
This tutorial is based on SciMLSensitivity.jl tutorial. Instead of using a classical NN architecture, here we will combine the NN with a symbolic expression from DynamicExpressions.jl (the symbolic engine behind SymbolicRegression.jl and PySR).
Here we will solve a classic optimal control problem with a universal differential equation. Let
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Package Imports ​
julia
using Lux, Boltz, ComponentArrays, OrdinaryDiffEqVerner, Optimization, OptimizationOptimJL,
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Solving Optimal Control Problems with Symbolic Universal Differential Equations ​
This tutorial is based on SciMLSensitivity.jl tutorial. Instead of using a classical NN architecture, here we will combine the NN with a symbolic expression from DynamicExpressions.jl (the symbolic engine behind SymbolicRegression.jl and PySR).
Here we will solve a classic optimal control problem with a universal differential equation. Let
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")),A("mjx-container",T,[(i(),a("svg",h,s[2]||(s[2]=[t('',1)]))),s[3]||(s[3]=A("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[A("mi",null,"u"),A("mo",{stretchy:"false"},"("),A("mi",null,"t"),A("mo",{stretchy:"false"},")")])],-1))]),s[5]||(s[5]=n(" such that the following is minimized:"))]),A("mjx-container",r,[(i(),a("svg",d,s[6]||(s[6]=[t('',1)]))),s[7]||(s[7]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mrow",{"data-mjx-texclass":"ORD"},[A("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"L")]),A("mo",{stretchy:"false"},"("),A("mi",null,"θ"),A("mo",{stretchy:"false"},")"),A("mo",null,"="),A("munder",null,[A("mo",{"data-mjx-texclass":"OP"},"∑"),A("mi",null,"i")]),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"4"),A("mo",null,"−"),A("mi",null,"x"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",null,"+"),A("mn",null,"2"),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mi",null,"x"),A("mi",{"data-mjx-alternate":"1"},"′"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",null,"+"),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mi",null,"u"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),A("p",null,[s[14]||(s[14]=n("where ")),A("mjx-container",o,[(i(),a("svg",k,s[8]||(s[8]=[A("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[A("g",{"data-mml-node":"math"},[A("g",{"data-mml-node":"mi"},[A("path",{"data-c":"1D456",d:"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),s[9]||(s[9]=A("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[A("mi",null,"i")])],-1))]),s[15]||(s[15]=n(" is measured on ")),A("mjx-container",m,[(i(),a("svg",g,s[10]||(s[10]=[t('',1)]))),s[11]||(s[11]=A("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[A("mo",{stretchy:"false"},"("),A("mn",null,"0"),A("mo",null,","),A("mn",null,"8"),A("mo",{stretchy:"false"},")")])],-1))]),s[16]||(s[16]=n(" at ")),A("mjx-container",E,[(i(),a("svg",c,s[12]||(s[12]=[t('',1)]))),s[13]||(s[13]=A("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[A("mn",null,"0.01")])],-1))]),s[17]||(s[17]=n(" intervals. To do this, we rewrite the ODE in first order form:"))]),A("mjx-container",y,[(i(),a("svg",C,s[18]||(s[18]=[t('',1)]))),s[19]||(s[19]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("msup",null,[A("mi",null,"x"),A("mi",{"data-mjx-alternate":"1"},"′")]),A("mo",null,"="),A("mi",null,"v")])],-1))]),A("mjx-container",u,[(i(),a("svg",f,s[20]||(s[20]=[t('',1)]))),s[21]||(s[21]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("msup",null,[A("mi",null,"v"),A("mi",{"data-mjx-alternate":"1"},"′")]),A("mo",null,"="),A("msup",null,[A("mi",null,"u"),A("mn",null,"3")]),A("mo",{stretchy:"false"},"("),A("mi",null,"t"),A("mo",{stretchy:"false"},")")])],-1))]),s[37]||(s[37]=A("p",null,"and thus",-1)),A("mjx-container",v,[(i(),a("svg",H,s[22]||(s[22]=[t('',1)]))),s[23]||(s[23]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mrow",{"data-mjx-texclass":"ORD"},[A("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"L")]),A("mo",{stretchy:"false"},"("),A("mi",null,"θ"),A("mo",{stretchy:"false"},")"),A("mo",null,"="),A("munder",null,[A("mo",{"data-mjx-texclass":"OP"},"∑"),A("mi",null,"i")]),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"4"),A("mo",null,"−"),A("mi",null,"x"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",null,"+"),A("mn",null,"2"),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mi",null,"v"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",null,"+"),A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mi",null,"u"),A("mo",{stretchy:"false"},"("),A("msub",null,[A("mi",null,"t"),A("mi",null,"i")]),A("mo",{stretchy:"false"},")"),A("msub",null,[A("mo",{"data-mjx-texclass":"ORD"},"∥"),A("mn",null,"2")]),A("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),A("p",null,[s[26]||(s[26]=n("is our loss function on the first order system. We thus choose a neural network form for ")),A("mjx-container",F,[(i(),a("svg",L,s[24]||(s[24]=[A("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[A("g",{"data-mml-node":"math"},[A("g",{"data-mml-node":"mi"},[A("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),s[25]||(s[25]=A("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[A("mi",null,"u")])],-1))]),s[27]||(s[27]=n(" and optimize the equation with respect to this loss. Note that we will first reduce control cost (the last term) by 10x in order to bump the network out of a local minimum. This looks like:"))]),s[38]||(s[38]=t(`
Package Imports ​
julia
using Lux, Boltz, ComponentArrays, OrdinaryDiffEqVerner, Optimization, OptimizationOptimJL,
       OptimizationOptimisers, SciMLSensitivity, Statistics, Printf, Random
 using DynamicExpressions, SymbolicRegression, MLJ, SymbolicUtils, Latexify
-using CairoMakie
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-   1539.4 ms  ✓ RandomNumbers
-   2136.6 ms  ✓ FastLapackInterface
-    638.4 ms  ✓ Scratch
-   1114.9 ms  ✓ Rmath_jll
-   1882.7 ms  ✓ KLU
-   1337.0 ms  ✓ oneTBB_jll
-   4881.4 ms  ✓ TimerOutputs
-   1908.9 ms  ✓ QuadGK
-   1056.0 ms  ✓ ResettableStacks
-   2298.9 ms  ✓ Enzyme_jll
-   2015.6 ms  ✓ IntelOpenMP_jll
-   2142.3 ms  ✓ HypergeometricFunctions
-    956.2 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsStructArraysExt
-   1288.4 ms  ✓ HostCPUFeatures
-  10943.2 ms  ✓ Krylov
-   4391.5 ms  ✓ SciMLJacobianOperators
-   2622.4 ms  ✓ DifferentiationInterface → DifferentiationInterfaceZygoteExt
-   8507.4 ms  ✓ Tracker
-   5426.4 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsZygoteExt
-   5997.9 ms  ✓ SciMLBase → SciMLBaseZygoteExt
-   3071.6 ms  ✓ ObjectFile
-    781.2 ms  ✓ OffsetArrays → OffsetArraysAdaptExt
-    960.2 ms  ✓ StaticArrayInterface → StaticArrayInterfaceOffsetArraysExt
-   1895.9 ms  ✓ Sparspak
-    742.7 ms  ✓ FunctionProperties
-  22077.3 ms  ✓ ArrayLayouts
-   1380.5 ms  ✓ Random123
-   1245.3 ms  ✓ Rmath
-   7276.4 ms  ✓ DiffEqCallbacks
-   1862.2 ms  ✓ DifferentiationInterface → DifferentiationInterfaceTrackerExt
-   1890.7 ms  ✓ Tracker → TrackerPDMatsExt
-   1730.3 ms  ✓ FastPower → FastPowerTrackerExt
-   1595.4 ms  ✓ ArrayInterface → ArrayInterfaceTrackerExt
-   1910.7 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsTrackerExt
-   3440.8 ms  ✓ Zygote → ZygoteTrackerExt
-  28679.1 ms  ✓ ReverseDiff
-  13504.5 ms  ✓ MKL_jll
-   1572.3 ms  ✓ ArrayLayouts → ArrayLayoutsSparseArraysExt
-  14792.6 ms  ✓ VectorizationBase
-   3208.6 ms  ✓ StatsFuns
-   4545.6 ms  ✓ DiffEqBase → DiffEqBaseTrackerExt
-   6164.9 ms  ✓ FastPower → FastPowerReverseDiffExt
-   7006.7 ms  ✓ DifferentiationInterface → DifferentiationInterfaceReverseDiffExt
-   5583.7 ms  ✓ ArrayInterface → ArrayInterfaceReverseDiffExt
-   9374.7 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsReverseDiffExt
-   4799.8 ms  ✓ LazyArrays
-   5961.5 ms  ✓ PreallocationTools → PreallocationToolsReverseDiffExt
-   1409.5 ms  ✓ StatsFuns → StatsFunsInverseFunctionsExt
-   1869.1 ms  ✓ SLEEFPirates
-   3062.0 ms  ✓ StatsFuns → StatsFunsChainRulesCoreExt
-   9366.1 ms  ✓ DiffEqBase → DiffEqBaseReverseDiffExt
-   2636.0 ms  ✓ LazyArrays → LazyArraysStaticArraysExt
-   7923.0 ms  ✓ Distributions
-   2699.9 ms  ✓ Distributions → DistributionsChainRulesCoreExt
-   3180.3 ms  ✓ Distributions → DistributionsTestExt
-   4012.2 ms  ✓ DiffEqBase → DiffEqBaseDistributionsExt
-  46203.5 ms  ✓ GPUCompiler
-   5219.4 ms  ✓ DiffEqNoiseProcess
-   6683.0 ms  ✓ DiffEqNoiseProcess → DiffEqNoiseProcessReverseDiffExt
-  34253.7 ms  ✓ LoopVectorization
-   1490.9 ms  ✓ LoopVectorization → SpecialFunctionsExt
-   1641.1 ms  ✓ LoopVectorization → ForwardDiffExt
-   3401.5 ms  ✓ TriangularSolve
-  13410.9 ms  ✓ RecursiveFactorization
-  33454.7 ms  ✓ LinearSolve
-   3325.6 ms  ✓ LinearSolve → LinearSolveRecursiveArrayToolsExt
-   3484.9 ms  ✓ LinearSolve → LinearSolveEnzymeExt
-   5075.3 ms  ✓ LinearSolve → LinearSolveKernelAbstractionsExt
- 212740.4 ms  ✓ Enzyme
-   9961.4 ms  ✓ Enzyme → EnzymeGPUArraysCoreExt
-  10286.3 ms  ✓ Enzyme → EnzymeSpecialFunctionsExt
-  10380.7 ms  ✓ Enzyme → EnzymeLogExpFunctionsExt
-  14476.3 ms  ✓ Enzyme → EnzymeStaticArraysExt
-  18207.1 ms  ✓ Enzyme → EnzymeChainRulesCoreExt
-   8949.5 ms  ✓ DifferentiationInterface → DifferentiationInterfaceEnzymeExt
-   8739.8 ms  ✓ FastPower → FastPowerEnzymeExt
-   8812.7 ms  ✓ QuadGK → QuadGKEnzymeExt
-   7650.9 ms  ✓ DiffEqBase → DiffEqBaseEnzymeExt
-  19093.8 ms  ✓ SciMLSensitivity
-  83 dependencies successfully precompiled in 321 seconds. 208 already precompiled.
-  1 dependency had output during precompilation:
-┌ MKL_jll
-│  \x1B[32m\x1B[1m Downloading\x1B[22m\x1B[39m artifact: IntelOpenMP
-└  
-Precompiling LuxLibSLEEFPiratesExt...
-   2116.8 ms  ✓ LuxLib → LuxLibSLEEFPiratesExt
-  1 dependency successfully precompiled in 2 seconds. 97 already precompiled.
-Precompiling LuxLibLoopVectorizationExt...
-   3528.7 ms  ✓ LuxLib → LuxLibLoopVectorizationExt
-  1 dependency successfully precompiled in 4 seconds. 105 already precompiled.
-Precompiling LuxLibEnzymeExt...
-   1113.5 ms  ✓ LuxLib → LuxLibEnzymeExt
-  1 dependency successfully precompiled in 1 seconds. 130 already precompiled.
+   1786.7 ms  ✓ DifferentiationInterface → DifferentiationInterfaceTrackerExt
+   4834.6 ms  ✓ SciMLJacobianOperators
+   3176.7 ms  ✓ DiffEqBase → DiffEqBaseDistributionsExt
+   6242.2 ms  ✓ DifferentiationInterface → DifferentiationInterfaceReverseDiffExt
+   4183.1 ms  ✓ DiffEqBase → DiffEqBaseTrackerExt
+  10767.7 ms  ✓ DifferentiationInterface → DifferentiationInterfaceEnzymeExt
+   7738.6 ms  ✓ DiffEqBase → DiffEqBaseReverseDiffExt
+   7501.2 ms  ✓ DiffEqCallbacks
+  14881.8 ms  ✓ DiffEqBase → DiffEqBaseEnzymeExt
+   6007.8 ms  ✓ DiffEqNoiseProcess
+   4274.9 ms  ✓ DiffEqNoiseProcess → DiffEqNoiseProcessReverseDiffExt
+  47009.1 ms  ✓ Zygote
+   3522.2 ms  ✓ DifferentiationInterface → DifferentiationInterfaceZygoteExt
+   3887.0 ms  ✓ Zygote → ZygoteTrackerExt
+   5977.6 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsZygoteExt
+   6538.0 ms  ✓ SciMLBase → SciMLBaseZygoteExt
+  46666.8 ms  ✓ LinearSolve
+   7062.2 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsReverseDiffExt
+   3319.5 ms  ✓ LinearSolve → LinearSolveEnzymeExt
+   3422.6 ms  ✓ LinearSolve → LinearSolveRecursiveArrayToolsExt
+   5051.1 ms  ✓ LinearSolve → LinearSolveKernelAbstractionsExt
+  19211.3 ms  ✓ SciMLSensitivity
+  22 dependencies successfully precompiled in 82 seconds. 269 already precompiled.
 Precompiling LuxEnzymeExt...
-   5931.9 ms  ✓ Lux → LuxEnzymeExt
+   5995.5 ms  ✓ Lux → LuxEnzymeExt
   1 dependency successfully precompiled in 6 seconds. 146 already precompiled.
 Precompiling OptimizationEnzymeExt...
-  11637.8 ms  ✓ OptimizationBase → OptimizationEnzymeExt
+  11812.7 ms  ✓ OptimizationBase → OptimizationEnzymeExt
   1 dependency successfully precompiled in 12 seconds. 109 already precompiled.
-Precompiling MLDataDevicesTrackerExt...
-   1005.5 ms  ✓ MLDataDevices → MLDataDevicesTrackerExt
-  1 dependency successfully precompiled in 1 seconds. 59 already precompiled.
-Precompiling LuxLibTrackerExt...
-    950.2 ms  ✓ LuxCore → LuxCoreArrayInterfaceTrackerExt
-   3019.0 ms  ✓ LuxLib → LuxLibTrackerExt
-  2 dependencies successfully precompiled in 3 seconds. 100 already precompiled.
 Precompiling LuxTrackerExt...
-   1839.3 ms  ✓ Lux → LuxTrackerExt
+   1855.7 ms  ✓ Lux → LuxTrackerExt
   1 dependency successfully precompiled in 2 seconds. 114 already precompiled.
 Precompiling BoltzTrackerExt...
-   2073.4 ms  ✓ Boltz → BoltzTrackerExt
+   2095.3 ms  ✓ Boltz → BoltzTrackerExt
   1 dependency successfully precompiled in 2 seconds. 128 already precompiled.
-Precompiling ComponentArraysTrackerExt...
-   1004.8 ms  ✓ ComponentArrays → ComponentArraysTrackerExt
-  1 dependency successfully precompiled in 1 seconds. 70 already precompiled.
-Precompiling MLDataDevicesReverseDiffExt...
-   2830.3 ms  ✓ MLDataDevices → MLDataDevicesReverseDiffExt
-  1 dependency successfully precompiled in 3 seconds. 49 already precompiled.
-Precompiling LuxLibReverseDiffExt...
-   2764.5 ms  ✓ LuxCore → LuxCoreArrayInterfaceReverseDiffExt
-   3503.6 ms  ✓ LuxLib → LuxLibReverseDiffExt
-  2 dependencies successfully precompiled in 4 seconds. 98 already precompiled.
 Precompiling BoltzReverseDiffExt...
-   3672.2 ms  ✓ Lux → LuxReverseDiffExt
-   3885.8 ms  ✓ Boltz → BoltzReverseDiffExt
+   3699.1 ms  ✓ Lux → LuxReverseDiffExt
+   3905.2 ms  ✓ Boltz → BoltzReverseDiffExt
   2 dependencies successfully precompiled in 4 seconds. 128 already precompiled.
-Precompiling ComponentArraysReverseDiffExt...
-   2818.4 ms  ✓ ComponentArrays → ComponentArraysReverseDiffExt
-  1 dependency successfully precompiled in 3 seconds. 57 already precompiled.
 Precompiling OptimizationReverseDiffExt...
-   2737.4 ms  ✓ OptimizationBase → OptimizationReverseDiffExt
+   2757.9 ms  ✓ OptimizationBase → OptimizationReverseDiffExt
   1 dependency successfully precompiled in 3 seconds. 130 already precompiled.
+Precompiling MLDataDevicesZygoteExt...
+   1406.1 ms  ✓ MLDataDevices → MLDataDevicesZygoteExt
+  1 dependency successfully precompiled in 2 seconds. 109 already precompiled.
+Precompiling LuxZygoteExt...
+   2483.1 ms  ✓ Lux → LuxZygoteExt
+  1 dependency successfully precompiled in 3 seconds. 166 already precompiled.
+Precompiling BoltzZygoteExt...
+   2401.1 ms  ✓ Boltz → BoltzZygoteExt
+  1 dependency successfully precompiled in 3 seconds. 168 already precompiled.
 Precompiling ComponentArraysZygoteExt...
-   1398.7 ms  ✓ ComponentArrays → ComponentArraysZygoteExt
-   1667.0 ms  ✓ ComponentArrays → ComponentArraysGPUArraysExt
-  2 dependencies successfully precompiled in 2 seconds. 116 already precompiled.
+   1380.6 ms  ✓ ComponentArrays → ComponentArraysZygoteExt
+  1 dependency successfully precompiled in 2 seconds. 117 already precompiled.
 Precompiling OptimizationZygoteExt...
-   1873.6 ms  ✓ OptimizationBase → OptimizationZygoteExt
+   1923.3 ms  ✓ OptimizationBase → OptimizationZygoteExt
   1 dependency successfully precompiled in 2 seconds. 160 already precompiled.
-Precompiling DynamicExpressionsOptimExt...
-   1163.8 ms  ✓ DynamicExpressions → DynamicExpressionsOptimExt
-  1 dependency successfully precompiled in 1 seconds. 77 already precompiled.
-Precompiling DynamicExpressionsLoopVectorizationExt...
-   3382.1 ms  ✓ DynamicExpressions → DynamicExpressionsLoopVectorizationExt
-  1 dependency successfully precompiled in 4 seconds. 49 already precompiled.
 Precompiling DynamicExpressionsZygoteExt...
-   1362.1 ms  ✓ DynamicExpressions → DynamicExpressionsZygoteExt
+   1404.6 ms  ✓ DynamicExpressions → DynamicExpressionsZygoteExt
   1 dependency successfully precompiled in 2 seconds. 106 already precompiled.
 Precompiling SymbolicRegression...
-    478.9 ms  ✓ Tricks
-    503.1 ms  ✓ ScientificTypesBase
-    538.6 ms  ✓ StatisticalTraits
-   1929.9 ms  ✓ LossFunctions
-    958.4 ms  ✓ MLJModelInterface
-   2325.1 ms  ✓ DynamicDiff
-   4045.2 ms  ✓ DynamicQuantities
-    605.1 ms  ✓ DynamicQuantities → DynamicQuantitiesLinearAlgebraExt
-  68194.5 ms  ✓ SymbolicRegression
-  9 dependencies successfully precompiled in 73 seconds. 100 already precompiled.
+  70209.6 ms  ✓ SymbolicRegression
+  1 dependency successfully precompiled in 70 seconds. 108 already precompiled.
 Precompiling LuxLossFunctionsExt...
-   1369.9 ms  ✓ Lux → LuxLossFunctionsExt
+   1425.1 ms  ✓ Lux → LuxLossFunctionsExt
   1 dependency successfully precompiled in 2 seconds. 110 already precompiled.
 Precompiling SymbolicRegressionEnzymeExt...
-  16133.6 ms  ✓ SymbolicRegression → SymbolicRegressionEnzymeExt
-  1 dependency successfully precompiled in 16 seconds. 129 already precompiled.
-Precompiling MLJ...
-    502.8 ms  ✓ LaTeXStrings
-    527.5 ms  ✓ SimpleBufferStream
-    579.1 ms  ✓ InvertedIndices
-    703.0 ms  ✓ BitFlags
-   1898.8 ms  ✓ Crayons
-    734.6 ms  ✓ StableRNGs
-   1637.0 ms  ✓ Combinatorics
-   1126.5 ms  ✓ ComputationalResources
-   1221.3 ms  ✓ ConcurrentUtilities
-   1369.8 ms  ✓ Distances
-   3935.9 ms  ✓ FixedPointNumbers
-   1893.0 ms  ✓ FeatureSelection
-    791.8 ms  ✓ EarlyStopping
-   1953.1 ms  ✓ MbedTLS
-    843.4 ms  ✓ RelocatableFolders
-   6313.5 ms  ✓ PrettyPrinting
-    947.1 ms  ✓ ExceptionUnwrapping
-   2212.7 ms  ✓ LearnAPI
-   4774.9 ms  ✓ CategoricalArrays
-   1684.1 ms  ✓ FilePathsBase
-    725.0 ms  ✓ Distances → DistancesChainRulesCoreExt
-   1176.0 ms  ✓ Distances → DistancesSparseArraysExt
-   1787.7 ms  ✓ LatinHypercubeSampling
-   3549.8 ms  ✓ OpenSSL
-   3899.5 ms  ✓ StringManipulation
-   1309.4 ms  ✓ CategoricalArrays → CategoricalArraysJSONExt
-   2401.1 ms  ✓ IterationControl
-   1367.4 ms  ✓ CategoricalArrays → CategoricalArraysRecipesBaseExt
-   1087.8 ms  ✓ FilePathsBase → FilePathsBaseMmapExt
-   2998.1 ms  ✓ ARFFFiles
-   4386.0 ms  ✓ ColorTypes
-   1967.4 ms  ✓ FilePathsBase → FilePathsBaseTestExt
-   9972.2 ms  ✓ StatisticalMeasuresBase
-  21064.9 ms  ✓ HTTP
-   2136.0 ms  ✓ MLFlowClient
-   3759.5 ms  ✓ OpenML
-  24683.0 ms  ✓ PrettyTables
-   2334.3 ms  ✓ ScientificTypes
-   2097.2 ms  ✓ CategoricalDistributions
-   4660.4 ms  ✓ MLJEnsembles
-   8620.5 ms  ✓ MLJBase
-  13854.8 ms  ✓ MLJModels
-   6729.0 ms  ✓ MLJTuning
-   6971.1 ms  ✓ MLJBalancing
-   7939.9 ms  ✓ MLJIteration
-   4437.3 ms  ✓ MLJFlow
-  22744.9 ms  ✓ StatisticalMeasures
-   2193.6 ms  ✓ StatisticalMeasures → ScientificTypesExt
-   2248.4 ms  ✓ MLJBase → DefaultMeasuresExt
-   5622.7 ms  ✓ MLJ
-  50 dependencies successfully precompiled in 73 seconds. 149 already precompiled.
-Precompiling LossFunctionsCategoricalArraysExt...
-    418.7 ms  ✓ LossFunctions → LossFunctionsCategoricalArraysExt
-  1 dependency successfully precompiled in 1 seconds. 12 already precompiled.
-Precompiling DynamicQuantitiesScientificTypesExt...
-   1327.5 ms  ✓ DynamicQuantities → DynamicQuantitiesScientificTypesExt
-  1 dependency successfully precompiled in 2 seconds. 74 already precompiled.
-Precompiling TransducersLazyArraysExt...
-   1042.3 ms  ✓ Transducers → TransducersLazyArraysExt
-  1 dependency successfully precompiled in 1 seconds. 48 already precompiled.
+  17149.6 ms  ✓ SymbolicRegression → SymbolicRegressionEnzymeExt
+  1 dependency successfully precompiled in 17 seconds. 129 already precompiled.
 Precompiling OptimizationMLUtilsExt...
-   1592.6 ms  ✓ OptimizationBase → OptimizationMLUtilsExt
+   1689.0 ms  ✓ OptimizationBase → OptimizationMLUtilsExt
   1 dependency successfully precompiled in 2 seconds. 151 already precompiled.
-Precompiling LossFunctionsExt...
-   2231.0 ms  ✓ StatisticalMeasures → LossFunctionsExt
-  1 dependency successfully precompiled in 2 seconds. 138 already precompiled.
-Precompiling SymbolicUtils...
-    496.5 ms  ✓ TermInterface
-    521.4 ms  ✓ Bijections
-    636.7 ms  ✓ WeakValueDicts
-    731.8 ms  ✓ Unityper
-   5228.8 ms  ✓ MutableArithmetics
-   2407.7 ms  ✓ MultivariatePolynomials
-   1505.7 ms  ✓ DynamicPolynomials
-  18053.0 ms  ✓ SymbolicUtils
-  8 dependencies successfully precompiled in 27 seconds. 71 already precompiled.
-Precompiling DynamicExpressionsSymbolicUtilsExt...
-   1743.2 ms  ✓ DynamicExpressions → DynamicExpressionsSymbolicUtilsExt
-  1 dependency successfully precompiled in 2 seconds. 83 already precompiled.
 Precompiling SymbolicRegressionSymbolicUtilsExt...
-   3592.3 ms  ✓ SymbolicRegression → SymbolicRegressionSymbolicUtilsExt
+   3869.7 ms  ✓ SymbolicRegression → SymbolicRegressionSymbolicUtilsExt
   1 dependency successfully precompiled in 4 seconds. 144 already precompiled.
-Precompiling SymbolicUtilsReverseDiffExt...
-   3627.5 ms  ✓ SymbolicUtils → SymbolicUtilsReverseDiffExt
-  1 dependency successfully precompiled in 4 seconds. 90 already precompiled.
-Precompiling Latexify...
-    974.8 ms  ✓ Format
-   3115.7 ms  ✓ Latexify
-  2 dependencies successfully precompiled in 4 seconds. 8 already precompiled.
-Precompiling SparseArraysExt...
-    708.5 ms  ✓ Latexify → SparseArraysExt
-  1 dependency successfully precompiled in 1 seconds. 31 already precompiled.
 Precompiling CairoMakie...
-    643.7 ms  ✓ GeoFormatTypes
-    723.5 ms  ✓ PaddedViews
-    772.0 ms  ✓ Contour
-    802.5 ms  ✓ Observables
-   1119.5 ms  ✓ Grisu
-    677.5 ms  ✓ Extents
-    633.7 ms  ✓ PolygonOps
-   1013.5 ms  ✓ IntervalSets
-    687.0 ms  ✓ StackViews
-    680.2 ms  ✓ RoundingEmulator
-    784.6 ms  ✓ IterTools
-    644.9 ms  ✓ LazyModules
-    788.2 ms  ✓ MappedArrays
-    499.0 ms  ✓ IndirectArrays
-    660.1 ms  ✓ RangeArrays
-    583.5 ms  ✓ TriplotBase
-   2423.3 ms  ✓ AdaptivePredicates
-    766.7 ms  ✓ Ratios
-    760.2 ms  ✓ TensorCore
-    890.9 ms  ✓ Inflate
-    970.4 ms  ✓ WoodburyMatrices
-    748.4 ms  ✓ SignedDistanceFields
-   1414.0 ms  ✓ FilePaths
-   1118.5 ms  ✓ Libffi_jll
-   2588.9 ms  ✓ UnicodeFun
-    983.1 ms  ✓ isoband_jll
-    982.4 ms  ✓ Libuuid_jll
-   1012.5 ms  ✓ LLVMOpenMP_jll
-   1086.6 ms  ✓ Imath_jll
-   1133.0 ms  ✓ CRlibm_jll
-    973.8 ms  ✓ Ogg_jll
-   1265.1 ms  ✓ JpegTurbo_jll
-    943.4 ms  ✓ x265_jll
-   1038.6 ms  ✓ x264_jll
-   1300.8 ms  ✓ FriBidi_jll
-   1401.9 ms  ✓ XML2_jll
-   1121.0 ms  ✓ Graphite2_jll
-   1324.4 ms  ✓ Xorg_libXau_jll
-   1252.8 ms  ✓ libpng_jll
-    976.5 ms  ✓ Giflib_jll
-   8091.0 ms  ✓ Colors
-   1085.6 ms  ✓ LAME_jll
-   1108.1 ms  ✓ EarCut_jll
-    886.0 ms  ✓ Xorg_libXdmcp_jll
-   1116.9 ms  ✓ libaom_jll
-    985.6 ms  ✓ LZO_jll
-   1103.4 ms  ✓ Opus_jll
-    867.2 ms  ✓ Xorg_xtrans_jll
-   1277.5 ms  ✓ Zstd_jll
-    929.9 ms  ✓ Libmount_jll
-    852.4 ms  ✓ Bzip2_jll
-   1220.5 ms  ✓ libfdk_aac_jll
-   1229.8 ms  ✓ LERC_jll
-    951.0 ms  ✓ XZ_jll
-    931.4 ms  ✓ Libgpg_error_jll
-    852.4 ms  ✓ Xorg_libpthread_stubs_jll
-   1205.9 ms  ✓ FFTW_jll
-    888.2 ms  ✓ Showoff
-   2578.4 ms  ✓ QOI
-   1745.3 ms  ✓ GeoInterface
-    739.1 ms  ✓ IntervalSets → IntervalSetsRandomExt
-    650.0 ms  ✓ IntervalSets → IntervalSetsStatisticsExt
-    799.0 ms  ✓ ConstructionBase → ConstructionBaseIntervalSetsExt
-   4475.7 ms  ✓ PkgVersion
-    793.2 ms  ✓ MosaicViews
-    760.7 ms  ✓ Ratios → RatiosFixedPointNumbersExt
-   1182.7 ms  ✓ AxisAlgorithms
-    781.0 ms  ✓ Isoband
-    976.2 ms  ✓ Pixman_jll
-   1255.0 ms  ✓ OpenEXR_jll
-   3959.7 ms  ✓ ColorVectorSpace
-   1488.2 ms  ✓ libvorbis_jll
-   1263.2 ms  ✓ Gettext_jll
-   1084.1 ms  ✓ libsixel_jll
-    971.0 ms  ✓ Graphics
-   1237.7 ms  ✓ Animations
-   4425.3 ms  ✓ IntervalArithmetic
-   1673.2 ms  ✓ ColorBrewer
-   1363.1 ms  ✓ FreeType2_jll
-   1311.3 ms  ✓ Libtiff_jll
-   1360.8 ms  ✓ Libgcrypt_jll
-   1640.2 ms  ✓ AxisArrays
-  15576.5 ms  ✓ SIMD
-   3210.0 ms  ✓ OpenEXR
-   4666.8 ms  ✓ Interpolations
-   1465.8 ms  ✓ ColorVectorSpace → SpecialFunctionsExt
-   8759.3 ms  ✓ ColorSchemes
-  41556.5 ms  ✓ Unitful
-   1009.8 ms  ✓ IntervalArithmetic → IntervalArithmeticIntervalSetsExt
-   1835.5 ms  ✓ Glib_jll
-   2539.9 ms  ✓ FreeType
-   1899.7 ms  ✓ Fontconfig_jll
-  23777.8 ms  ✓ FFTW
-   1423.7 ms  ✓ XSLT_jll
-  27184.1 ms  ✓ GeometryBasics
-  10088.2 ms  ✓ ExactPredicates
-   1339.0 ms  ✓ Unitful → InverseFunctionsUnitfulExt
-   1005.3 ms  ✓ Unitful → ConstructionBaseUnitfulExt
-   1931.1 ms  ✓ Interpolations → InterpolationsUnitfulExt
-   3195.3 ms  ✓ KernelDensity
-   2452.5 ms  ✓ Xorg_libxcb_jll
-   2921.5 ms  ✓ Packing
-  42820.0 ms  ✓ ImageCore
-  29113.1 ms  ✓ Automa
-  25649.5 ms  ✓ PlotUtils
-   2670.3 ms  ✓ ShaderAbstractions
-  16311.5 ms  ✓ GridLayoutBase
-   4760.3 ms  ✓ FreeTypeAbstraction
-   1540.5 ms  ✓ Xorg_libX11_jll
-   9640.9 ms  ✓ MakieCore
-   4272.8 ms  ✓ ImageBase
-   6048.1 ms  ✓ JpegTurbo
-  10563.5 ms  ✓ DelaunayTriangulation
-   8385.2 ms  ✓ PNGFiles
-   1260.0 ms  ✓ Xorg_libXrender_jll
-   1184.1 ms  ✓ Xorg_libXext_jll
-   6671.7 ms  ✓ Sixel
-   1298.2 ms  ✓ Cairo_jll
-   1619.2 ms  ✓ Libglvnd_jll
-   3256.0 ms  ✓ ImageAxes
-   1242.2 ms  ✓ HarfBuzz_jll
-   1424.5 ms  ✓ libwebp_jll
-   1444.3 ms  ✓ libass_jll
-   2059.2 ms  ✓ ImageMetadata
-   1277.5 ms  ✓ Pango_jll
-   1574.4 ms  ✓ FFMPEG_jll
-   4032.9 ms  ✓ WebP
-   3445.3 ms  ✓ Netpbm
-   2292.1 ms  ✓ Cairo
-  14010.5 ms  ✓ MathTeXEngine
-  86745.5 ms  ✓ TiffImages
-   1152.2 ms  ✓ ImageIO
- 137885.2 ms  ✓ Makie
-  88186.8 ms  ✓ CairoMakie
-  134 dependencies successfully precompiled in 366 seconds. 136 already precompiled.
+  12839.0 ms  ✓ GeometryBasics
+   2012.1 ms  ✓ Packing
+   2517.0 ms  ✓ ShaderAbstractions
+   3476.3 ms  ✓ FreeTypeAbstraction
+   7241.8 ms  ✓ MakieCore
+  10402.1 ms  ✓ GridLayoutBase
+  11443.7 ms  ✓ MathTeXEngine
+  65781.6 ms  ✓ TiffImages
+ 140851.2 ms  ✓ Makie
+  87241.3 ms  ✓ CairoMakie
+  10 dependencies successfully precompiled in 295 seconds. 260 already precompiled.
 Precompiling SparseMatrixColoringsColorsExt...
-    824.8 ms  ✓ SparseMatrixColorings → SparseMatrixColoringsColorsExt
+    842.2 ms  ✓ SparseMatrixColorings → SparseMatrixColoringsColorsExt
   1 dependency successfully precompiled in 1 seconds. 29 already precompiled.
 Precompiling ZygoteColorsExt...
-   1601.6 ms  ✓ Zygote → ZygoteColorsExt
+   1601.8 ms  ✓ Zygote → ZygoteColorsExt
   1 dependency successfully precompiled in 2 seconds. 105 already precompiled.
-Precompiling IntervalSetsExt...
-    906.7 ms  ✓ Accessors → IntervalSetsExt
-  1 dependency successfully precompiled in 1 seconds. 15 already precompiled.
-Precompiling IntervalSetsRecipesBaseExt...
-    551.3 ms  ✓ IntervalSets → IntervalSetsRecipesBaseExt
-  1 dependency successfully precompiled in 1 seconds. 9 already precompiled.
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Helper Functions ​
julia
function plot_dynamics(sol, us, ts)
     fig = Figure()
     ax = CairoMakie.Axis(fig[1, 1]; xlabel=L"t")
@@ -695,26 +301,26 @@ import{_ as e,c as A,a2 as n,j as s,a as i,o as t}from"./chunks/framework.BI0fNM
 r = report(mach)
 best_eq = [r.equations[1][r.best_idx[1]], r.equations[2][r.best_idx[2]],
     r.equations[3][r.best_idx[3]], r.equations[4][r.best_idx[4]]]
4-element Vector{DynamicExpressions.ExpressionModule.Expression{Float64, DynamicExpressions.NodeModule.Node{Float64}, @NamedTuple{operators::DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}, variable_names::Vector{String}}}}:
- (x3 * -0.502328657412731) - (((x2 * 0.061929614395115545) - 0.004810464255583077) * ((x1 / -0.22037385096229364) + (((x3 - (0.5953134353667464 / 1.2553627279851565)) * x2) - -1.8057907515075073)))
- (x2 * -0.3486243198551685) + (-0.13520854135894056 - (x4 * (x4 * ((x2 * -24.09890185117278) + 1.4390086486201723))))
- (1.2880839661497687 - ((1.3654078758771315 - (((x1 * (x1 + -0.6434378096334555)) * x2) + ((x3 - x1) * (3.4536618215595043 - ((x3 * -6.634586090196312) + (-0.01956117602530656 / x2)))))) * x2)) - x1
- ((((x2 - (x4 * 2.149357629081865)) * 15.551691733525809) + 3.173903690663501) * (x2 - 0.007497421660196726)) - x3
Let's see the expressions that SymbolicRegression.jl found. In case you were wondering, these expressions are not hardcoded, it is live updated from the output of the code above using Latexify.jl and the integration of SymbolicUtils.jl with DynamicExpressions.jl.
`,35)),s("mjx-container",b,[(t(),A("svg",I,a[28]||(a[28]=[n('',1)]))),a[29]||(a[29]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"x"),s("mn",null,"3"),s("mo",null,"⋅"),s("mo",null,"−"),s("mn",null,"0.50233"),s("mo",null,"−"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"⋅"),s("mn",null,"0.06193"),s("mo",null,"−"),s("mn",null,"0.0048105"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mfrac",null,[s("mrow",null,[s("mi",null,"x"),s("mn",null,"1")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"0.22037")])]),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"3"),s("mo",null,"−"),s("mn",null,"0.47422"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"+"),s("mn",null,"1.8058"),s("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),s("mjx-container",L,[(t(),A("svg",H,a[30]||(a[30]=[n('',1)]))),a[31]||(a[31]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"⋅"),s("mo",null,"−"),s("mn",null,"0.34862"),s("mo",null,"−"),s("mn",null,"0.13521"),s("mo",null,"−"),s("mi",null,"x"),s("mn",null,"4"),s("mo",null,"⋅"),s("mi",null,"x"),s("mn",null,"4"),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"⋅"),s("mo",null,"−"),s("mn",null,"24.099"),s("mo",null,"+"),s("mn",null,"1.439"),s("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),s("mjx-container",D,[(t(),A("svg",w,a[32]||(a[32]=[n('',1)]))),a[33]||(a[33]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mn",null,"1.2881"),s("mo",null,"−"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mn",null,"1.3654"),s("mo",null,"−"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"1"),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"1"),s("mo",null,"−"),s("mn",null,"0.64344"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"3"),s("mo",null,"−"),s("mi",null,"x"),s("mn",null,"1"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mn",null,"3.4537"),s("mo",null,"−"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"3"),s("mo",null,"⋅"),s("mo",null,"−"),s("mn",null,"6.6346"),s("mo",null,"+"),s("mfrac",null,[s("mrow",null,[s("mo",null,"−"),s("mn",null,"0.019561")]),s("mrow",null,[s("mi",null,"x"),s("mn",null,"2")])]),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"−"),s("mi",null,"x"),s("mn",null,"1")])],-1))]),s("mjx-container",M,[(t(),A("svg",B,a[34]||(a[34]=[n('',1)]))),a[35]||(a[35]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"−"),s("mi",null,"x"),s("mn",null,"4"),s("mo",null,"⋅"),s("mn",null,"2.1494"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mn",null,"15.552"),s("mo",null,"+"),s("mn",null,"3.1739"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mi",null,"x"),s("mn",null,"2"),s("mo",null,"−"),s("mn",null,"0.0074974"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"−"),s("mi",null,"x"),s("mn",null,"3")])],-1))]),a[39]||(a[39]=n(`
Combining the Neural Network with the Symbolic Expression ​
Now that we have the symbolic expression, we can combine it with the neural network to solve the optimal control problem. but we do need to perform some finetuning.
julia
hybrid_mlp = Chain(Dense(1 => 4, gelu),
+ ((x3 * -0.5811118531294479) - (((((x3 - x4) - x1) * (0.2185594485440549 - x2)) - ((x2 + x1) - x3)) * -0.11739316213798096)) - -0.008676987740604923
+ ((x2 + -0.05275118849143026) * ((-0.35079797158000764 - x4) - ((x2 * -6.409404555880295) * (x1 - x3)))) - 0.15372685484771115
+ 1.2880796636898961 - ((x2 + (x3 - (x4 * 1.401536058411267))) * ((x3 - (((x1 * (x2 * -1.5751052415073825)) + ((x1 * 1.079758987582803) - x3)) + x4)) + 1.3672555162301787))
+ (-0.02240665211743358 - x4) + ((((x4 + (x4 - x2)) * -16.278980820211682) - -3.149178297043239) * x2)
Let's see the expressions that SymbolicRegression.jl found. In case you were wondering, these expressions are not hardcoded, it is live updated from the output of the code above using Latexify.jl and the integration of SymbolicUtils.jl with DynamicExpressions.jl.
`,35)),A("mjx-container",V,[(i(),a("svg",b,s[28]||(s[28]=[t('',1)]))),s[29]||(s[29]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mi",null,"x"),A("mn",null,"3"),A("mo",null,"⋅"),A("mo",null,"−"),A("mn",null,"0.58111"),A("mo",null,"−"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"3"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"1"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mn",null,"0.21856"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"2"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"−"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"2"),A("mo",null,"+"),A("mi",null,"x"),A("mn",null,"1"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"3"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mo",null,"−"),A("mn",null,"0.11739"),A("mo",null,"+"),A("mn",null,"0.008677")])],-1))]),A("mjx-container",I,[(i(),a("svg",x,s[30]||(s[30]=[t('',1)]))),s[31]||(s[31]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"2"),A("mo",null,"−"),A("mn",null,"0.052751"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mo",null,"−"),A("mn",null,"0.3508"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"2"),A("mo",null,"⋅"),A("mo",null,"−"),A("mn",null,"6.4094"),A("mo",null,"⋅"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"1"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"3"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"−"),A("mn",null,"0.15373")])],-1))]),A("mjx-container",w,[(i(),a("svg",Z,s[32]||(s[32]=[t('',1)]))),s[33]||(s[33]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mn",null,"1.2881"),A("mo",null,"−"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"2"),A("mo",null,"+"),A("mi",null,"x"),A("mn",null,"3"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"⋅"),A("mn",null,"1.4015"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"3"),A("mo",null,"−"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"1"),A("mo",null,"⋅"),A("mi",null,"x"),A("mn",null,"2"),A("mo",null,"⋅"),A("mo",null,"−"),A("mn",null,"1.5751"),A("mo",null,"+"),A("mi",null,"x"),A("mn",null,"1"),A("mo",null,"⋅"),A("mn",null,"1.0798"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"3"),A("mo",null,"+"),A("mi",null,"x"),A("mn",null,"4"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"+"),A("mn",null,"1.3673"),A("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),A("mjx-container",B,[(i(),a("svg",M,s[34]||(s[34]=[t('',1)]))),s[35]||(s[35]=A("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[A("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[A("mo",null,"−"),A("mn",null,"0.022407"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"+"),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mrow",{"data-mjx-texclass":"INNER"},[A("mo",{"data-mjx-texclass":"OPEN"},"("),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"+"),A("mi",null,"x"),A("mn",null,"4"),A("mo",null,"−"),A("mi",null,"x"),A("mn",null,"2"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mo",null,"−"),A("mn",null,"16.279"),A("mo",null,"+"),A("mn",null,"3.1492"),A("mo",{"data-mjx-texclass":"CLOSE"},")")]),A("mo",null,"⋅"),A("mi",null,"x"),A("mn",null,"2")])],-1))]),s[39]||(s[39]=t(`
Combining the Neural Network with the Symbolic Expression ​
Now that we have the symbolic expression, we can combine it with the neural network to solve the optimal control problem. but we do need to perform some finetuning.
julia
hybrid_mlp = Chain(Dense(1 => 4, gelu),
     Layers.DynamicExpressionsLayer(OperatorEnum(; binary_operators=[+, -, *, /]), best_eq),
     Dense(4 => 1))
Chain(
     layer_1 = Dense(1 => 4, gelu),      # 8 parameters
     layer_2 = DynamicExpressionsLayer(
         chain = Chain(
             layer_1 = Parallel(
-                layer_1 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (x3 * -0.502328657412731) - (((x2 * 0.061929614395115545) - 0.004810464255583077) * ((x1 / -0.22037385096229364) + (((x3 - (0.5953134353667464 / 1.2553627279851565)) * x2) - -1.8057907515075073))); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 7 parameters
-                layer_2 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (x2 * -0.3486243198551685) + (-0.13520854135894056 - (x4 * (x4 * ((x2 * -24.09890185117278) + 1.4390086486201723)))); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
-                layer_3 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (1.2880839661497687 - ((1.3654078758771315 - (((x1 * (x1 + -0.6434378096334555)) * x2) + ((x3 - x1) * (3.4536618215595043 - ((x3 * -6.634586090196312) + (-0.01956117602530656 / x2)))))) * x2)) - x1; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 6 parameters
-                layer_4 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((((x2 - (x4 * 2.149357629081865)) * 15.551691733525809) + 3.173903690663501) * (x2 - 0.007497421660196726)) - x3; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_1 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((x3 * -0.5811118531294479) - (((((x3 - x4) - x1) * (0.2185594485440549 - x2)) - ((x2 + x1) - x3)) * -0.11739316213798096)) - -0.008676987740604923; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_2 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((x2 + -0.05275118849143026) * ((-0.35079797158000764 - x4) - ((x2 * -6.409404555880295) * (x1 - x3)))) - 0.15372685484771115; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_3 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), 1.2880796636898961 - ((x2 + (x3 - (x4 * 1.401536058411267))) * ((x3 - (((x1 * (x2 * -1.5751052415073825)) + ((x1 * 1.079758987582803) - x3)) + x4)) + 1.3672555162301787)); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 5 parameters
+                layer_4 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (-0.02240665211743358 - x4) + ((((x4 + (x4 - x2)) * -16.278980820211682) - -3.149178297043239) * x2); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 3 parameters
             ),
             layer_2 = WrappedFunction(stack1),
         ),
     ),
     layer_3 = Dense(4 => 1),            # 5 parameters
-)         # Total: 34 parameters,
+)         # Total: 29 parameters,
           #        plus 0 states.
There you have it! It is that easy to take the fitted Symbolic Expression and combine it with a neural network. Let's see how it performs before fintetuning.
julia
hybrid_ude = construct_ude(hybrid_mlp, Vern9(); abstol=1e-10, reltol=1e-10);
We want to reuse the trained neural network parameters, so we will copy them over to the new model
julia
st = Lux.initialstates(rng, hybrid_ude)
 ps = (;
     mlp=(; layer_1=trained_ude.ps.mlp.layer_1,
@@ -723,7 +329,7 @@ import{_ as e,c as A,a2 as n,j as s,a as i,o as t}from"./chunks/framework.BI0fNM
 ps = ComponentArray(ps)
 
 sol, us = hybrid_ude(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), ps, st)[1];
-plot_dynamics(sol, us, 0.0:0.01:8.0)
Now that does perform well! But we could finetune this model very easily. We will skip that part on CI, but you can do it by using the same training code as above.
Appendix ​
julia
using InteractiveUtils
+plot_dynamics(sol, us, 0.0:0.01:8.0)
Now that does perform well! But we could finetune this model very easily. We will skip that part on CI, but you can do it by using the same training code as above.
Appendix ​
julia
using InteractiveUtils
 InteractiveUtils.versioninfo()
 
 if @isdefined(MLDataDevices)
@@ -750,4 +356,4 @@ import{_ as e,c as A,a2 as n,j as s,a as i,o as t}from"./chunks/framework.BI0fNM
   JULIA_NUM_THREADS = 1
   JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
   JULIA_PKG_PRECOMPILE_AUTO = 0
-  JULIA_DEBUG = Literate
This page was generated using Literate.jl.
`,15))])}const R=e(l,[["render",Z]]);export{O as __pageData,R as default};
+  JULIA_DEBUG = Literate
This page was generated using Literate.jl.
`,15))])}const O=e(l,[["render",D]]);export{X as __pageData,O as default};
diff --git a/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.lean.js b/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.lean.js
new file mode 100644
index 0000000..4296081
--- /dev/null
+++ b/previews/PR98/assets/tutorials_2_SymbolicOptimalControl.md.CNDr_s0o.lean.js
@@ -0,0 +1 @@
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diff --git a/previews/PR98/hashmap.json b/previews/PR98/hashmap.json
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diff --git a/previews/PR98/index.html b/previews/PR98/index.html
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Boltz.jl Docs
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How to Install Boltz.jl? ​
Its easy to install Boltz.jl. Since Boltz.jl is registered in the Julia General registry, you can simply run the following command in the Julia REPL:
julia
julia> using Pkg
 julia> Pkg.add("Boltz")
If you want to use the latest unreleased version of Boltz.jl, you can run the following command: (in most cases the released version will be same as the version on github)
julia
julia> using Pkg
-julia> Pkg.add(url="https://github.com/LuxDL/Boltz.jl")
Want GPU Support? ​
Install the following package(s):
julia
using Pkg
+julia> Pkg.add(url="https://github.com/LuxDL/Boltz.jl")
Want GPU Support? ​
Install the following package(s):
julia
using Pkg
 Pkg.add("LuxCUDA")
 # or
 Pkg.add(["CUDA", "cuDNN"])
julia
using Pkg
 Pkg.add("AMDGPU")
julia
using Pkg
 Pkg.add("Metal")
julia
using Pkg
 Pkg.add("oneAPI")

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   JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
   JULIA_PKG_PRECOMPILE_AUTO = 0
   JULIA_DEBUG = Literate

This page was generated using Literate.jl.

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Machine Learning

Solving Optimal Control Problems with Symbolic Universal Differential Equations ​

This tutorial is based on SciMLSensitivity.jl tutorial. Instead of using a classical NN architecture, here we will combine the NN with a symbolic expression from DynamicExpressions.jl (the symbolic engine behind SymbolicRegression.jl and PySR).

Here we will solve a classic optimal control problem with a universal differential equation. Let

x′′=u3(t)

where we want to optimize our controller u(t) such that the following is minimized:

L(θ)=∑i(∥4−x(ti)∥2+2∥x′(ti)∥2+∥u(ti)∥2)

where i is measured on (0,8) at 0.01 intervals. To do this, we rewrite the ODE in first order form:

x′=vv′=u3(t)

and thus

L(θ)=∑i(∥4−x(ti)∥2+2∥v(ti)∥2+∥u(ti)∥2)

is our loss function on the first order system. We thus choose a neural network form for u and optimize the equation with respect to this loss. Note that we will first reduce control cost (the last term) by 10x in order to bump the network out of a local minimum. This looks like:

Package Imports ​

julia
using Lux, Boltz, ComponentArrays, OrdinaryDiffEqVerner, Optimization, OptimizationOptimJL,
       OptimizationOptimisers, SciMLSensitivity, Statistics, Printf, Random
 using DynamicExpressions, SymbolicRegression, MLJ, SymbolicUtils, Latexify
-using CairoMakie
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-   5583.7 ms  ✓ ArrayInterface → ArrayInterfaceReverseDiffExt
-   9374.7 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsReverseDiffExt
-   4799.8 ms  ✓ LazyArrays
-   5961.5 ms  ✓ PreallocationTools → PreallocationToolsReverseDiffExt
-   1409.5 ms  ✓ StatsFuns → StatsFunsInverseFunctionsExt
-   1869.1 ms  ✓ SLEEFPirates
-   3062.0 ms  ✓ StatsFuns → StatsFunsChainRulesCoreExt
-   9366.1 ms  ✓ DiffEqBase → DiffEqBaseReverseDiffExt
-   2636.0 ms  ✓ LazyArrays → LazyArraysStaticArraysExt
-   7923.0 ms  ✓ Distributions
-   2699.9 ms  ✓ Distributions → DistributionsChainRulesCoreExt
-   3180.3 ms  ✓ Distributions → DistributionsTestExt
-   4012.2 ms  ✓ DiffEqBase → DiffEqBaseDistributionsExt
-  46203.5 ms  ✓ GPUCompiler
-   5219.4 ms  ✓ DiffEqNoiseProcess
-   6683.0 ms  ✓ DiffEqNoiseProcess → DiffEqNoiseProcessReverseDiffExt
-  34253.7 ms  ✓ LoopVectorization
-   1490.9 ms  ✓ LoopVectorization → SpecialFunctionsExt
-   1641.1 ms  ✓ LoopVectorization → ForwardDiffExt
-   3401.5 ms  ✓ TriangularSolve
-  13410.9 ms  ✓ RecursiveFactorization
-  33454.7 ms  ✓ LinearSolve
-   3325.6 ms  ✓ LinearSolve → LinearSolveRecursiveArrayToolsExt
-   3484.9 ms  ✓ LinearSolve → LinearSolveEnzymeExt
-   5075.3 ms  ✓ LinearSolve → LinearSolveKernelAbstractionsExt
- 212740.4 ms  ✓ Enzyme
-   9961.4 ms  ✓ Enzyme → EnzymeGPUArraysCoreExt
-  10286.3 ms  ✓ Enzyme → EnzymeSpecialFunctionsExt
-  10380.7 ms  ✓ Enzyme → EnzymeLogExpFunctionsExt
-  14476.3 ms  ✓ Enzyme → EnzymeStaticArraysExt
-  18207.1 ms  ✓ Enzyme → EnzymeChainRulesCoreExt
-   8949.5 ms  ✓ DifferentiationInterface → DifferentiationInterfaceEnzymeExt
-   8739.8 ms  ✓ FastPower → FastPowerEnzymeExt
-   8812.7 ms  ✓ QuadGK → QuadGKEnzymeExt
-   7650.9 ms  ✓ DiffEqBase → DiffEqBaseEnzymeExt
-  19093.8 ms  ✓ SciMLSensitivity
-  83 dependencies successfully precompiled in 321 seconds. 208 already precompiled.
-  1 dependency had output during precompilation:
-┌ MKL_jll
-│   Downloading artifact: IntelOpenMP
-└  
-Precompiling LuxLibSLEEFPiratesExt...
-   2116.8 ms  ✓ LuxLib → LuxLibSLEEFPiratesExt
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-Precompiling LuxLibLoopVectorizationExt...
-   3528.7 ms  ✓ LuxLib → LuxLibLoopVectorizationExt
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-Precompiling LuxLibEnzymeExt...
-   1113.5 ms  ✓ LuxLib → LuxLibEnzymeExt
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+   1786.7 ms  ✓ DifferentiationInterface → DifferentiationInterfaceTrackerExt
+   4834.6 ms  ✓ SciMLJacobianOperators
+   3176.7 ms  ✓ DiffEqBase → DiffEqBaseDistributionsExt
+   6242.2 ms  ✓ DifferentiationInterface → DifferentiationInterfaceReverseDiffExt
+   4183.1 ms  ✓ DiffEqBase → DiffEqBaseTrackerExt
+  10767.7 ms  ✓ DifferentiationInterface → DifferentiationInterfaceEnzymeExt
+   7738.6 ms  ✓ DiffEqBase → DiffEqBaseReverseDiffExt
+   7501.2 ms  ✓ DiffEqCallbacks
+  14881.8 ms  ✓ DiffEqBase → DiffEqBaseEnzymeExt
+   6007.8 ms  ✓ DiffEqNoiseProcess
+   4274.9 ms  ✓ DiffEqNoiseProcess → DiffEqNoiseProcessReverseDiffExt
+  47009.1 ms  ✓ Zygote
+   3522.2 ms  ✓ DifferentiationInterface → DifferentiationInterfaceZygoteExt
+   3887.0 ms  ✓ Zygote → ZygoteTrackerExt
+   5977.6 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsZygoteExt
+   6538.0 ms  ✓ SciMLBase → SciMLBaseZygoteExt
+  46666.8 ms  ✓ LinearSolve
+   7062.2 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsReverseDiffExt
+   3319.5 ms  ✓ LinearSolve → LinearSolveEnzymeExt
+   3422.6 ms  ✓ LinearSolve → LinearSolveRecursiveArrayToolsExt
+   5051.1 ms  ✓ LinearSolve → LinearSolveKernelAbstractionsExt
+  19211.3 ms  ✓ SciMLSensitivity
+  22 dependencies successfully precompiled in 82 seconds. 269 already precompiled.
 Precompiling LuxEnzymeExt...
-   5931.9 ms  ✓ Lux → LuxEnzymeExt
+   5995.5 ms  ✓ Lux → LuxEnzymeExt
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 Precompiling OptimizationEnzymeExt...
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+  11812.7 ms  ✓ OptimizationBase → OptimizationEnzymeExt
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-Precompiling MLDataDevicesTrackerExt...
-   1005.5 ms  ✓ MLDataDevices → MLDataDevicesTrackerExt
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-   3019.0 ms  ✓ LuxLib → LuxLibTrackerExt
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 Precompiling LuxTrackerExt...
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+   1855.7 ms  ✓ Lux → LuxTrackerExt
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 Precompiling BoltzTrackerExt...
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+   2095.3 ms  ✓ Boltz → BoltzTrackerExt
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-   1004.8 ms  ✓ ComponentArrays → ComponentArraysTrackerExt
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-   2764.5 ms  ✓ LuxCore → LuxCoreArrayInterfaceReverseDiffExt
-   3503.6 ms  ✓ LuxLib → LuxLibReverseDiffExt
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 Precompiling BoltzReverseDiffExt...
-   3672.2 ms  ✓ Lux → LuxReverseDiffExt
-   3885.8 ms  ✓ Boltz → BoltzReverseDiffExt
+   3699.1 ms  ✓ Lux → LuxReverseDiffExt
+   3905.2 ms  ✓ Boltz → BoltzReverseDiffExt
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-   2818.4 ms  ✓ ComponentArrays → ComponentArraysReverseDiffExt
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 Precompiling OptimizationReverseDiffExt...
-   2737.4 ms  ✓ OptimizationBase → OptimizationReverseDiffExt
+   2757.9 ms  ✓ OptimizationBase → OptimizationReverseDiffExt
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+   1406.1 ms  ✓ MLDataDevices → MLDataDevicesZygoteExt
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+Precompiling LuxZygoteExt...
+   2483.1 ms  ✓ Lux → LuxZygoteExt
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+   2401.1 ms  ✓ Boltz → BoltzZygoteExt
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 Precompiling ComponentArraysZygoteExt...
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 Precompiling OptimizationZygoteExt...
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 Precompiling DynamicExpressionsZygoteExt...
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 Precompiling SymbolicRegression...
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-    958.4 ms  ✓ MLJModelInterface
-   2325.1 ms  ✓ DynamicDiff
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 Precompiling LuxLossFunctionsExt...
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 Precompiling SymbolicRegressionEnzymeExt...
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 Precompiling OptimizationMLUtilsExt...
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 Precompiling SymbolicRegressionSymbolicUtilsExt...
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 Precompiling CairoMakie...
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-    852.4 ms  ✓ Xorg_libpthread_stubs_jll
-   1205.9 ms  ✓ FFTW_jll
-    888.2 ms  ✓ Showoff
-   2578.4 ms  ✓ QOI
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-    739.1 ms  ✓ IntervalSets → IntervalSetsRandomExt
-    650.0 ms  ✓ IntervalSets → IntervalSetsStatisticsExt
-    799.0 ms  ✓ ConstructionBase → ConstructionBaseIntervalSetsExt
-   4475.7 ms  ✓ PkgVersion
-    793.2 ms  ✓ MosaicViews
-    760.7 ms  ✓ Ratios → RatiosFixedPointNumbersExt
-   1182.7 ms  ✓ AxisAlgorithms
-    781.0 ms  ✓ Isoband
-    976.2 ms  ✓ Pixman_jll
-   1255.0 ms  ✓ OpenEXR_jll
-   3959.7 ms  ✓ ColorVectorSpace
-   1488.2 ms  ✓ libvorbis_jll
-   1263.2 ms  ✓ Gettext_jll
-   1084.1 ms  ✓ libsixel_jll
-    971.0 ms  ✓ Graphics
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-   1673.2 ms  ✓ ColorBrewer
-   1363.1 ms  ✓ FreeType2_jll
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-   1360.8 ms  ✓ Libgcrypt_jll
-   1640.2 ms  ✓ AxisArrays
-  15576.5 ms  ✓ SIMD
-   3210.0 ms  ✓ OpenEXR
-   4666.8 ms  ✓ Interpolations
-   1465.8 ms  ✓ ColorVectorSpace → SpecialFunctionsExt
-   8759.3 ms  ✓ ColorSchemes
-  41556.5 ms  ✓ Unitful
-   1009.8 ms  ✓ IntervalArithmetic → IntervalArithmeticIntervalSetsExt
-   1835.5 ms  ✓ Glib_jll
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-   1899.7 ms  ✓ Fontconfig_jll
-  23777.8 ms  ✓ FFTW
-   1423.7 ms  ✓ XSLT_jll
-  27184.1 ms  ✓ GeometryBasics
-  10088.2 ms  ✓ ExactPredicates
-   1339.0 ms  ✓ Unitful → InverseFunctionsUnitfulExt
-   1005.3 ms  ✓ Unitful → ConstructionBaseUnitfulExt
-   1931.1 ms  ✓ Interpolations → InterpolationsUnitfulExt
-   3195.3 ms  ✓ KernelDensity
-   2452.5 ms  ✓ Xorg_libxcb_jll
-   2921.5 ms  ✓ Packing
-  42820.0 ms  ✓ ImageCore
-  29113.1 ms  ✓ Automa
-  25649.5 ms  ✓ PlotUtils
-   2670.3 ms  ✓ ShaderAbstractions
-  16311.5 ms  ✓ GridLayoutBase
-   4760.3 ms  ✓ FreeTypeAbstraction
-   1540.5 ms  ✓ Xorg_libX11_jll
-   9640.9 ms  ✓ MakieCore
-   4272.8 ms  ✓ ImageBase
-   6048.1 ms  ✓ JpegTurbo
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-   1260.0 ms  ✓ Xorg_libXrender_jll
-   1184.1 ms  ✓ Xorg_libXext_jll
-   6671.7 ms  ✓ Sixel
-   1298.2 ms  ✓ Cairo_jll
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-   2059.2 ms  ✓ ImageMetadata
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-   1574.4 ms  ✓ FFMPEG_jll
-   4032.9 ms  ✓ WebP
-   3445.3 ms  ✓ Netpbm
-   2292.1 ms  ✓ Cairo
-  14010.5 ms  ✓ MathTeXEngine
-  86745.5 ms  ✓ TiffImages
-   1152.2 ms  ✓ ImageIO
- 137885.2 ms  ✓ Makie
-  88186.8 ms  ✓ CairoMakie
-  134 dependencies successfully precompiled in 366 seconds. 136 already precompiled.
+  12839.0 ms  ✓ GeometryBasics
+   2012.1 ms  ✓ Packing
+   2517.0 ms  ✓ ShaderAbstractions
+   3476.3 ms  ✓ FreeTypeAbstraction
+   7241.8 ms  ✓ MakieCore
+  10402.1 ms  ✓ GridLayoutBase
+  11443.7 ms  ✓ MathTeXEngine
+  65781.6 ms  ✓ TiffImages
+ 140851.2 ms  ✓ Makie
+  87241.3 ms  ✓ CairoMakie
+  10 dependencies successfully precompiled in 295 seconds. 260 already precompiled.
 Precompiling SparseMatrixColoringsColorsExt...
-    824.8 ms  ✓ SparseMatrixColorings → SparseMatrixColoringsColorsExt
+    842.2 ms  ✓ SparseMatrixColorings → SparseMatrixColoringsColorsExt
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 Precompiling ZygoteColorsExt...
-   1601.6 ms  ✓ Zygote → ZygoteColorsExt
+   1601.8 ms  ✓ Zygote → ZygoteColorsExt
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-    906.7 ms  ✓ Accessors → IntervalSetsExt
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-    551.3 ms  ✓ IntervalSets → IntervalSetsRecipesBaseExt
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 Precompiling DiffEqBaseUnitfulExt...
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+   1353.9 ms  ✓ DiffEqBase → DiffEqBaseUnitfulExt
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-   1009.0 ms  ✓ DynamicQuantities → DynamicQuantitiesUnitfulExt
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-    473.4 ms  ✓ IntervalArithmetic → IntervalArithmeticDiffRulesExt
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-    762.9 ms  ✓ IntervalArithmetic → IntervalArithmeticRecipesBaseExt
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 Precompiling SciMLBaseMakieExt...
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+   7014.0 ms  ✓ SciMLBase → SciMLBaseMakieExt
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Helper Functions ​

julia
function plot_dynamics(sol, us, ts)
     fig = Figure()
     ax = CairoMakie.Axis(fig[1, 1]; xlabel=L"t")
@@ -722,26 +328,26 @@
 r = report(mach)
 best_eq = [r.equations[1][r.best_idx[1]], r.equations[2][r.best_idx[2]],
     r.equations[3][r.best_idx[3]], r.equations[4][r.best_idx[4]]]
4-element Vector{DynamicExpressions.ExpressionModule.Expression{Float64, DynamicExpressions.NodeModule.Node{Float64}, @NamedTuple{operators::DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}, variable_names::Vector{String}}}}:
- (x3 * -0.502328657412731) - (((x2 * 0.061929614395115545) - 0.004810464255583077) * ((x1 / -0.22037385096229364) + (((x3 - (0.5953134353667464 / 1.2553627279851565)) * x2) - -1.8057907515075073)))
- (x2 * -0.3486243198551685) + (-0.13520854135894056 - (x4 * (x4 * ((x2 * -24.09890185117278) + 1.4390086486201723))))
- (1.2880839661497687 - ((1.3654078758771315 - (((x1 * (x1 + -0.6434378096334555)) * x2) + ((x3 - x1) * (3.4536618215595043 - ((x3 * -6.634586090196312) + (-0.01956117602530656 / x2)))))) * x2)) - x1
- ((((x2 - (x4 * 2.149357629081865)) * 15.551691733525809) + 3.173903690663501) * (x2 - 0.007497421660196726)) - x3

Let's see the expressions that SymbolicRegression.jl found. In case you were wondering, these expressions are not hardcoded, it is live updated from the output of the code above using Latexify.jl and the integration of SymbolicUtils.jl with DynamicExpressions.jl.

x3⋅−0.50233−(x2⋅0.06193−0.0048105)⋅(x1−0.22037+(x3−0.47422)⋅x2+1.8058)x2⋅−0.34862−0.13521−x4⋅x4⋅(x2⋅−24.099+1.439)1.2881−(1.3654−(x1⋅(x1−0.64344)⋅x2+(x3−x1)⋅(3.4537−(x3⋅−6.6346+−0.019561x2))))⋅x2−x1((x2−x4⋅2.1494)⋅15.552+3.1739)⋅(x2−0.0074974)−x3

Combining the Neural Network with the Symbolic Expression ​

Now that we have the symbolic expression, we can combine it with the neural network to solve the optimal control problem. but we do need to perform some finetuning.

julia
hybrid_mlp = Chain(Dense(1 => 4, gelu),
+ ((x3 * -0.5811118531294479) - (((((x3 - x4) - x1) * (0.2185594485440549 - x2)) - ((x2 + x1) - x3)) * -0.11739316213798096)) - -0.008676987740604923
+ ((x2 + -0.05275118849143026) * ((-0.35079797158000764 - x4) - ((x2 * -6.409404555880295) * (x1 - x3)))) - 0.15372685484771115
+ 1.2880796636898961 - ((x2 + (x3 - (x4 * 1.401536058411267))) * ((x3 - (((x1 * (x2 * -1.5751052415073825)) + ((x1 * 1.079758987582803) - x3)) + x4)) + 1.3672555162301787))
+ (-0.02240665211743358 - x4) + ((((x4 + (x4 - x2)) * -16.278980820211682) - -3.149178297043239) * x2)

Let's see the expressions that SymbolicRegression.jl found. In case you were wondering, these expressions are not hardcoded, it is live updated from the output of the code above using Latexify.jl and the integration of SymbolicUtils.jl with DynamicExpressions.jl.

x3⋅−0.58111−((x3−x4−x1)⋅(0.21856−x2)−(x2+x1−x3))⋅−0.11739+0.008677(x2−0.052751)⋅(−0.3508−x4−x2⋅−6.4094⋅(x1−x3))−0.153731.2881−(x2+x3−x4⋅1.4015)⋅(x3−(x1⋅x2⋅−1.5751+x1⋅1.0798−x3+x4)+1.3673)−0.022407−x4+((x4+x4−x2)⋅−16.279+3.1492)⋅x2

Combining the Neural Network with the Symbolic Expression ​

Now that we have the symbolic expression, we can combine it with the neural network to solve the optimal control problem. but we do need to perform some finetuning.

julia
hybrid_mlp = Chain(Dense(1 => 4, gelu),
     Layers.DynamicExpressionsLayer(OperatorEnum(; binary_operators=[+, -, *, /]), best_eq),
     Dense(4 => 1))
Chain(
     layer_1 = Dense(1 => 4, gelu),      # 8 parameters
     layer_2 = DynamicExpressionsLayer(
         chain = Chain(
             layer_1 = Parallel(
-                layer_1 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (x3 * -0.502328657412731) - (((x2 * 0.061929614395115545) - 0.004810464255583077) * ((x1 / -0.22037385096229364) + (((x3 - (0.5953134353667464 / 1.2553627279851565)) * x2) - -1.8057907515075073))); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 7 parameters
-                layer_2 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (x2 * -0.3486243198551685) + (-0.13520854135894056 - (x4 * (x4 * ((x2 * -24.09890185117278) + 1.4390086486201723)))); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
-                layer_3 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (1.2880839661497687 - ((1.3654078758771315 - (((x1 * (x1 + -0.6434378096334555)) * x2) + ((x3 - x1) * (3.4536618215595043 - ((x3 * -6.634586090196312) + (-0.01956117602530656 / x2)))))) * x2)) - x1; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 6 parameters
-                layer_4 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((((x2 - (x4 * 2.149357629081865)) * 15.551691733525809) + 3.173903690663501) * (x2 - 0.007497421660196726)) - x3; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_1 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((x3 * -0.5811118531294479) - (((((x3 - x4) - x1) * (0.2185594485440549 - x2)) - ((x2 + x1) - x3)) * -0.11739316213798096)) - -0.008676987740604923; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_2 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), ((x2 + -0.05275118849143026) * ((-0.35079797158000764 - x4) - ((x2 * -6.409404555880295) * (x1 - x3)))) - 0.15372685484771115; eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 4 parameters
+                layer_3 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), 1.2880796636898961 - ((x2 + (x3 - (x4 * 1.401536058411267))) * ((x3 - (((x1 * (x2 * -1.5751052415073825)) + ((x1 * 1.079758987582803) - x3)) + x4)) + 1.3672555162301787)); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 5 parameters
+                layer_4 = InternalDynamicExpressionWrapper(DynamicExpressions.OperatorEnumModule.OperatorEnum{Tuple{typeof(+), typeof(-), typeof(*), typeof(/)}, Tuple{}}((+, -, *, /), ()), (-0.02240665211743358 - x4) + ((((x4 + (x4 - x2)) * -16.278980820211682) - -3.149178297043239) * x2); eval_options=(turbo = Val{false}(), bumper = Val{false}())),  # 3 parameters
             ),
             layer_2 = WrappedFunction(stack1),
         ),
     ),
     layer_3 = Dense(4 => 1),            # 5 parameters
-)         # Total: 34 parameters,
+)         # Total: 29 parameters,
           #        plus 0 states.

There you have it! It is that easy to take the fitted Symbolic Expression and combine it with a neural network. Let's see how it performs before fintetuning.

julia
hybrid_ude = construct_ude(hybrid_mlp, Vern9(); abstol=1e-10, reltol=1e-10);

We want to reuse the trained neural network parameters, so we will copy them over to the new model

julia
st = Lux.initialstates(rng, hybrid_ude)
 ps = (;
     mlp=(; layer_1=trained_ude.ps.mlp.layer_1,
@@ -750,7 +356,7 @@
 ps = ComponentArray(ps)
 
 sol, us = hybrid_ude(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), ps, st)[1];
-plot_dynamics(sol, us, 0.0:0.01:8.0)

Now that does perform well! But we could finetune this model very easily. We will skip that part on CI, but you can do it by using the same training code as above.

Appendix ​

julia
using InteractiveUtils
+plot_dynamics(sol, us, 0.0:0.01:8.0)

Now that does perform well! But we could finetune this model very easily. We will skip that part on CI, but you can do it by using the same training code as above.

Appendix ​

julia
using InteractiveUtils
 InteractiveUtils.versioninfo()
 
 if @isdefined(MLDataDevices)
@@ -778,7 +384,7 @@
   JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
   JULIA_PKG_PRECOMPILE_AUTO = 0
   JULIA_DEBUG = Literate

This page was generated using Literate.jl.

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