diff --git a/previews/PR98/404.html b/previews/PR98/404.html new file mode 100644 index 0000000..93878e7 --- /dev/null +++ b/previews/PR98/404.html @@ -0,0 +1,29 @@ + + + + + + 404 | Boltz.jl Docs + + + + + + + + + + + + + + + + + + +
+ + + + \ No newline at end of file diff --git a/previews/PR98/api/basis.html b/previews/PR98/api/basis.html new file mode 100644 index 0000000..8d782b4 --- /dev/null +++ b/previews/PR98/api/basis.html @@ -0,0 +1,32 @@ + + + + + + Boltz.Basis API Reference | 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

+ + + + \ No newline at end of file diff --git a/previews/PR98/api/index.html b/previews/PR98/api/index.html new file mode 100644 index 0000000..521f3c4 --- /dev/null +++ b/previews/PR98/api/index.html @@ -0,0 +1,32 @@ + + + + + + API Reference | 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...;
+    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]);
+
+julia> x1 = Node(; feature=1);
+
+julia> x2 = Node(; feature=2);
+
+julia> expr_1 = x1 * cos(x2 - 3.2)
+x1 * cos(x2 - 3.2)
+
+julia> expr_2 = x2 - x1 * x2 + 2.5 - 1.0 * x1
+((x2 - (x1 * x2)) + 2.5) - (1.0 * x1)
+
+julia> layer = Layers.DynamicExpressionsLayer(operators, expr_1, expr_2);
+
+julia> ps, st = Lux.setup(Random.default_rng(), layer)
+((layer_1 = (layer_1 = (params = Float32[3.2],), layer_2 = (params = Float32[2.5, 1.0],)), layer_2 = NamedTuple()), (layer_1 = (layer_1 = NamedTuple(), layer_2 = NamedTuple()), layer_2 = NamedTuple()))
+
+julia> x = [1.0f0 2.0f0 3.0f0
+            4.0f0 5.0f0 6.0f0]
+2×3 Matrix{Float32}:
+ 1.0  2.0  3.0
+ 4.0  5.0  6.0
+
+julia> layer(x, ps, st)[1] ≈ Float32[0.6967068 -0.4544041 -2.8266668; 1.5 -4.5 -12.5]
+true
+
+julia> ∂x, ∂ps, _ = Zygote.gradient(Base.Fix1(sum, abs2) ∘ first ∘ layer, x, ps, st);
+
+julia> ∂x ≈ Float32[-14.0292 54.206482 180.32669; -0.9995737 10.7700815 55.6814]
+true
+
+julia> ∂ps.layer_1.layer_1.params ≈ Float32[-6.451908]
+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,
+    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])
+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},
+    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.

+ + + + \ No newline at end of file diff --git a/previews/PR98/api/private.html b/previews/PR98/api/private.html new file mode 100644 index 0000000..35620e4 --- /dev/null +++ b/previews/PR98/api/private.html @@ -0,0 +1,32 @@ + + + + + + Private API | 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 new file mode 100644 index 0000000..7ee112b --- /dev/null +++ b/previews/PR98/api/vision.html @@ -0,0 +1,32 @@ + + + + + + Computer Vision Models (Vision API) | 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.

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API Reference ​

Accelerate ⚡ your ML research using pre-built Deep Learning Models with Lux.

Index ​

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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

',3))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"Boltz.Layers.ConvNormActivation",href:"#Boltz.Layers.ConvNormActivation"},[s("span",{class:"jlbinding"},"Boltz.Layers.ConvNormActivation")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[5]||(i[5]=t(`
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

Keyword Arguments

source

`,7))]),s("details",T,[s("summary",null,[i[6]||(i[6]=s("a",{id:"Boltz.Layers.DynamicExpressionsLayer",href:"#Boltz.Layers.DynamicExpressionsLayer"},[s("span",{class:"jlbinding"},"Boltz.Layers.DynamicExpressionsLayer")],-1)),i[7]||(i[7]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[8]||(i[8]=t(`
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

Keyword Arguments

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]);
+
+julia> x1 = Node(; feature=1);
+
+julia> x2 = Node(; feature=2);
+
+julia> expr_1 = x1 * cos(x2 - 3.2)
+x1 * cos(x2 - 3.2)
+
+julia> expr_2 = x2 - x1 * x2 + 2.5 - 1.0 * x1
+((x2 - (x1 * x2)) + 2.5) - (1.0 * x1)
+
+julia> layer = Layers.DynamicExpressionsLayer(operators, expr_1, expr_2);
+
+julia> ps, st = Lux.setup(Random.default_rng(), layer)
+((layer_1 = (layer_1 = (params = Float32[3.2],), layer_2 = (params = Float32[2.5, 1.0],)), layer_2 = NamedTuple()), (layer_1 = (layer_1 = NamedTuple(), layer_2 = NamedTuple()), layer_2 = NamedTuple()))
+
+julia> x = [1.0f0 2.0f0 3.0f0
+            4.0f0 5.0f0 6.0f0]
+2×3 Matrix{Float32}:
+ 1.0  2.0  3.0
+ 4.0  5.0  6.0
+
+julia> layer(x, ps, st)[1] ≈ Float32[0.6967068 -0.4544041 -2.8266668; 1.5 -4.5 -12.5]
+true
+
+julia> ∂x, ∂ps, _ = Zygote.gradient(Base.Fix1(sum, abs2) ∘ first ∘ layer, x, ps, st);
+
+julia> ∂x ≈ Float32[-14.0292 54.206482 180.32669; -0.9995737 10.7700815 55.6814]
+true
+
+julia> ∂ps.layer_1.layer_1.params ≈ Float32[-6.451908]
+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

Keyword Arguments

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

Keyword Arguments

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

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

Keyword Arguments

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

Inputs

Returns

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

Keyword Arguments

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)
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Keyword Arguments

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

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julia
ViPosEmbedding(embedding_size, number_patches; init = randn32)

Positional embedding layer used by many vision transformer-like models.

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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

Keyword Arguments

source

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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

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Bibliography ​

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Type-stable and faster version of MLUtils.chunk.

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

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second_dim_mean(x)

Computes the mean of x along dimension 2.

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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.

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AlexNet(; kwargs...)

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

Keyword Arguments

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VGG(imsize; config, inchannels, batchnorm = false, nclasses, fcsize, dropout)

Create a VGG model (Simonyan, 2014).

Arguments

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

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

Arguments

Keyword Arguments

source

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Creates a Vision Transformer model with the specified configuration.

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

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

Arguments

Keyword Arguments

source

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julia
DenseNet(depth::Int; pretrained::Bool=false)

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

Arguments

Keyword Arguments

source

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julia
GoogLeNet(; pretrained::Bool=false)

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

Keyword Arguments

source

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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

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julia
ResNet(depth::Int; pretrained::Bool=false)

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

Arguments

Keyword Arguments

source

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

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

Arguments

Keyword Arguments

source

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julia
SqueezeNet(; pretrained::Bool=false)

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

Keyword Arguments

source

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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 ​

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t=this.opt.caseSensitive?"":"i",n=this.opt.caseSensitive?["aàáảãạăằắẳẵặâầấẩẫậäåāą","AÀÁẢÃẠĂẰẮẲẴẶÂẦẤẨẪẬÄÅĀĄ","cçćč","CÇĆČ","dđď","DĐĎ","eèéẻẽẹêềếểễệëěēę","EÈÉẺẼẸÊỀẾỂỄỆËĚĒĘ","iìíỉĩịîïī","IÌÍỈĨỊÎÏĪ","lł","LŁ","nñňń","NÑŇŃ","oòóỏõọôồốổỗộơởỡớờợöøō","OÒÓỎÕỌÔỒỐỔỖỘƠỞỠỚỜỢÖØŌ","rř","RŘ","sšśșş","SŠŚȘŞ","tťțţ","TŤȚŢ","uùúủũụưừứửữựûüůū","UÙÚỦŨỤƯỪỨỬỮỰÛÜŮŪ","yýỳỷỹỵÿ","YÝỲỶỸỴŸ","zžżź","ZŽŻŹ"]:["aàáảãạăằắẳẵặâầấẩẫậäåāąAÀÁẢÃẠĂẰẮẲẴẶÂẦẤẨẪẬÄÅĀĄ","cçćčCÇĆČ","dđďDĐĎ","eèéẻẽẹêềếểễệëěēęEÈÉẺẼẸÊỀẾỂỄỆËĚĒĘ","iìíỉĩịîïīIÌÍỈĨỊÎÏĪ","lłLŁ","nñňńNÑŇŃ","oòóỏõọôồốổỗộơởỡớờợöøōOÒÓỎÕỌÔỒỐỔỖỘƠỞỠỚỜỢÖØŌ","rřRŘ","sšśșşSŠŚȘŞ","tťțţTŤȚŢ","uùúủũụưừứửữựûüůūUÙÚỦŨỤƯỪỨỬỮỰÛÜŮŪ","yýỳỷỹỵÿYÝỲỶỸỴŸ","zžżźZŽŻŹ"];let s=[];return e.split("").forEach(r=>{n.every(i=>{if(i.indexOf(r)!==-1){if(s.indexOf(i)>-1)return!1;e=e.replace(new RegExp(`[${i}]`,`gm${t}`),`[${i}]`),s.push(i)}return!0})}),e}createMergedBlanksRegExp(e){return e.replace(/[\s]+/gmi,"[\\s]+")}createAccuracyRegExp(e){const t="!\"#$%&'()*+,-./:;<=>?@[\\]^_`{|}~¡¿";let 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n=document.createDocumentFragment();for(;e.firstChild;)n.appendChild(e.removeChild(e.firstChild));t.replaceChild(n,e),this.ie?this.normalizeTextNode(t):t.normalize()}normalizeTextNode(e){if(e){if(e.nodeType===3)for(;e.nextSibling&&e.nextSibling.nodeType===3;)e.nodeValue+=e.nextSibling.nodeValue,e.parentNode.removeChild(e.nextSibling);else this.normalizeTextNode(e.firstChild);this.normalizeTextNode(e.nextSibling)}}markRegExp(e,t){this.opt=t,this.log(`Searching with expression "${e}"`);let n=0,s="wrapMatches";const r=i=>{n++,this.opt.each(i)};this.opt.acrossElements&&(s="wrapMatchesAcrossElements"),this[s](e,this.opt.ignoreGroups,(i,o)=>this.opt.filter(o,i,n),r,()=>{n===0&&this.opt.noMatch(e),this.opt.done(n)})}mark(e,t){this.opt=t;let n=0,s="wrapMatches";const{keywords:r,length:i}=this.getSeparatedKeywords(typeof e=="string"?[e]:e),o=this.opt.caseSensitive?"":"i",l=c=>{let f=new RegExp(this.createRegExp(c),`gm${o}`),g=0;this.log(`Searching with expression 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s===void 0&&n.set(D,s=t()),s}values(){return new De(this,St)}[Symbol.iterator](){return this.entries()}static from(e){const t=new X;for(const[n,s]of e)t.set(n,s);return t}static fromObject(e){return X.from(Object.entries(e))}}const Re=(a,e,t=[])=>{if(e.length===0||a==null)return[a,t];for(const n of a.keys())if(n!==D&&e.startsWith(n))return t.push([a,n]),Re(a.get(n),e.slice(n.length),t);return t.push([a,e]),Re(void 0,"",t)},Ke=(a,e)=>{if(e.length===0||a==null)return a;for(const t of a.keys())if(t!==D&&e.startsWith(t))return Ke(a.get(t),e.slice(t.length))},ze=(a,e)=>{const t=e.length;e:for(let n=0;a&&n{const[t,n]=Re(a,e);if(t!==void 0){if(t.delete(D),t.size===0)Tt(n);else if(t.size===1){const[s,r]=t.entries().next().value;It(n,s,r)}}},Tt=a=>{if(a.length===0)return;const[e,t]=Ue(a);if(e.delete(t),e.size===0)Tt(a.slice(0,-1));else 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Map,this._dirtCount=0,this._currentVacuum=null,this._enqueuedVacuum=null,this._enqueuedVacuumConditions=qe,this.addFields(this._options.fields)}add(e){const{extractField:t,tokenize:n,processTerm:s,fields:r,idField:i}=this._options,o=t(e,i);if(o==null)throw new Error(`MiniSearch: document does not have ID field "${i}"`);if(this._idToShortId.has(o))throw new Error(`MiniSearch: duplicate ID ${o}`);const l=this.addDocumentId(o);this.saveStoredFields(l,e);for(const c of r){const f=t(e,c);if(f==null)continue;const g=n(f.toString(),c),h=this._fieldIds[c],b=new Set(g).size;this.addFieldLength(l,h,this._documentCount-1,b);for(const y of g){const x=s(y,c);if(Array.isArray(x))for(const w of x)this.addTerm(h,l,w);else x&&this.addTerm(h,l,x)}}}addAll(e){for(const t of e)this.add(t)}addAllAsync(e,t={}){const{chunkSize:n=10}=t,s={chunk:[],promise:Promise.resolve()},{chunk:r,promise:i}=e.reduce(({chunk:o,promise:l},c,f)=>(o.push(c),(f+1)%n===0?{chunk:[],promise:l.then(()=>new 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e)this.discard(n)}finally{this._options.autoVacuum=t}this.maybeAutoVacuum()}replace(e){const{idField:t,extractField:n}=this._options,s=n(e,t);this.discard(s),this.add(e)}vacuum(e={}){return this.conditionalVacuum(e)}conditionalVacuum(e,t){return this._currentVacuum?(this._enqueuedVacuumConditions=this._enqueuedVacuumConditions&&t,this._enqueuedVacuum!=null?this._enqueuedVacuum:(this._enqueuedVacuum=this._currentVacuum.then(()=>{const n=this._enqueuedVacuumConditions;return this._enqueuedVacuumConditions=qe,this.performVacuuming(e,n)}),this._enqueuedVacuum)):this.vacuumConditionsMet(t)===!1?Promise.resolve():(this._currentVacuum=this.performVacuuming(e),this._currentVacuum)}performVacuuming(e,t){return ke(this,void 0,void 0,function*(){const n=this._dirtCount;if(this.vacuumConditionsMet(t)){const s=e.batchSize||Je.batchSize,r=e.batchWait||Je.batchWait;let i=1;for(const[o,l]of this._index){for(const[c,f]of l)for(const[g]of 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n)s[r]=Object.fromEntries(i);e.push([t,s])}return{documentCount:this._documentCount,nextId:this._nextId,documentIds:Object.fromEntries(this._documentIds),fieldIds:this._fieldIds,fieldLength:Object.fromEntries(this._fieldLength),averageFieldLength:this._avgFieldLength,storedFields:Object.fromEntries(this._storedFields),dirtCount:this._dirtCount,index:e,serializationVersion:2}}termResults(e,t,n,s,r,i,o,l,c=new Map){if(r==null)return c;for(const f of Object.keys(i)){const g=i[f],h=this._fieldIds[f],b=r.get(h);if(b==null)continue;let y=b.size;const x=this._avgFieldLength[h];for(const w of b.keys()){if(!this._documentIds.has(w)){this.removeTerm(h,w,t),y-=1;continue}const R=o?o(this._documentIds.get(w),t,this._storedFields.get(w)):1;if(!R)continue;const A=b.get(w),J=this._fieldLength.get(w)[h],Q=Un(A,y,this._documentCount,J,x,l),W=n*s*g*R*Q,V=c.get(w);if(V){V.score+=W,Qn(V.terms,e);const $=Pe(V.match,t);$?$.push(f):V.match[t]=[f]}else c.set(w,{score:W,terms:[e],match:{[t]:[f]}})}}return 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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
+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")
`,8)]))}const E=s(e,[["render",n]]);export{g as __pageData,E as default}; diff --git a/previews/PR98/assets/index.md.ac-RXHi6.lean.js b/previews/PR98/assets/index.md.ac-RXHi6.lean.js new file mode 100644 index 0000000..1cedb03 --- /dev/null +++ b/previews/PR98/assets/index.md.ac-RXHi6.lean.js @@ -0,0 +1 @@ +import{_ as s,c as a,a2 as t,o as l}from"./chunks/framework.BI0fNMXE.js";const g=JSON.parse('{"title":"","description":"","frontmatter":{"layout":"home","hero":{"name":"Boltz.jl ⚡ Docs","text":"Pre-built Deep Learning Models in Julia","tagline":"Accelerate ⚡ your ML research using pre-built Deep Learning Models with Lux","actions":[{"theme":"brand","text":"Lux.jl Docs","link":"https://lux.csail.mit.edu/"},{"theme":"alt","text":"Tutorials 📚","link":"/tutorials/1_GettingStarted"},{"theme":"alt","text":"Vision Models 👀","link":"/api/vision"},{"theme":"alt","text":"Layers API 🧩","link":"/api/layers"},{"theme":"alt","text":"View on 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#c2c2c4;--vp-c-divider: #e2e2e3;--vp-c-gutter: #e2e2e3}.dark{--vp-c-border: #3c3f44;--vp-c-divider: #2e2e32;--vp-c-gutter: #000000}:root{--vp-c-text-1: #3c3c43;--vp-c-text-2: #67676c;--vp-c-text-3: #929295}.dark{--vp-c-text-1: #dfdfd6;--vp-c-text-2: #98989f;--vp-c-text-3: #6a6a71}:root{--vp-c-default-1: var(--vp-c-gray-1);--vp-c-default-2: var(--vp-c-gray-2);--vp-c-default-3: var(--vp-c-gray-3);--vp-c-default-soft: var(--vp-c-gray-soft);--vp-c-brand-1: var(--vp-c-indigo-1);--vp-c-brand-2: var(--vp-c-indigo-2);--vp-c-brand-3: var(--vp-c-indigo-3);--vp-c-brand-soft: var(--vp-c-indigo-soft);--vp-c-brand: var(--vp-c-brand-1);--vp-c-tip-1: var(--vp-c-brand-1);--vp-c-tip-2: var(--vp-c-brand-2);--vp-c-tip-3: var(--vp-c-brand-3);--vp-c-tip-soft: var(--vp-c-brand-soft);--vp-c-note-1: var(--vp-c-brand-1);--vp-c-note-2: var(--vp-c-brand-2);--vp-c-note-3: var(--vp-c-brand-3);--vp-c-note-soft: var(--vp-c-brand-soft);--vp-c-success-1: var(--vp-c-green-1);--vp-c-success-2: var(--vp-c-green-2);--vp-c-success-3: var(--vp-c-green-3);--vp-c-success-soft: var(--vp-c-green-soft);--vp-c-important-1: var(--vp-c-purple-1);--vp-c-important-2: var(--vp-c-purple-2);--vp-c-important-3: var(--vp-c-purple-3);--vp-c-important-soft: var(--vp-c-purple-soft);--vp-c-warning-1: var(--vp-c-yellow-1);--vp-c-warning-2: var(--vp-c-yellow-2);--vp-c-warning-3: var(--vp-c-yellow-3);--vp-c-warning-soft: var(--vp-c-yellow-soft);--vp-c-danger-1: var(--vp-c-red-1);--vp-c-danger-2: var(--vp-c-red-2);--vp-c-danger-3: var(--vp-c-red-3);--vp-c-danger-soft: var(--vp-c-red-soft);--vp-c-caution-1: var(--vp-c-red-1);--vp-c-caution-2: var(--vp-c-red-2);--vp-c-caution-3: var(--vp-c-red-3);--vp-c-caution-soft: var(--vp-c-red-soft)}:root{--vp-font-family-base: "Inter", ui-sans-serif, system-ui, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";--vp-font-family-mono: ui-monospace, "Menlo", "Monaco", "Consolas", "Liberation Mono", "Courier New", monospace;font-optical-sizing:auto}:root:where(:lang(zh)){--vp-font-family-base: "Punctuation SC", "Inter", ui-sans-serif, system-ui, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"}:root{--vp-shadow-1: 0 1px 2px rgba(0, 0, 0, .04), 0 1px 2px rgba(0, 0, 0, .06);--vp-shadow-2: 0 3px 12px rgba(0, 0, 0, .07), 0 1px 4px rgba(0, 0, 0, .07);--vp-shadow-3: 0 12px 32px rgba(0, 0, 0, .1), 0 2px 6px rgba(0, 0, 0, .08);--vp-shadow-4: 0 14px 44px rgba(0, 0, 0, .12), 0 3px 9px rgba(0, 0, 0, .12);--vp-shadow-5: 0 18px 56px rgba(0, 0, 0, .16), 0 4px 12px rgba(0, 0, 0, .16)}:root{--vp-z-index-footer: 10;--vp-z-index-local-nav: 20;--vp-z-index-nav: 30;--vp-z-index-layout-top: 40;--vp-z-index-backdrop: 50;--vp-z-index-sidebar: 60}@media (min-width: 960px){:root{--vp-z-index-sidebar: 25}}:root{--vp-layout-max-width: 1440px}:root{--vp-header-anchor-symbol: "#"}:root{--vp-code-line-height: 1.7;--vp-code-font-size: .875em;--vp-code-color: var(--vp-c-brand-1);--vp-code-link-color: var(--vp-c-brand-1);--vp-code-link-hover-color: var(--vp-c-brand-2);--vp-code-bg: var(--vp-c-default-soft);--vp-code-block-color: var(--vp-c-text-2);--vp-code-block-bg: var(--vp-c-bg-alt);--vp-code-block-divider-color: var(--vp-c-gutter);--vp-code-lang-color: var(--vp-c-text-3);--vp-code-line-highlight-color: var(--vp-c-default-soft);--vp-code-line-number-color: var(--vp-c-text-3);--vp-code-line-diff-add-color: var(--vp-c-success-soft);--vp-code-line-diff-add-symbol-color: var(--vp-c-success-1);--vp-code-line-diff-remove-color: var(--vp-c-danger-soft);--vp-code-line-diff-remove-symbol-color: var(--vp-c-danger-1);--vp-code-line-warning-color: var(--vp-c-warning-soft);--vp-code-line-error-color: var(--vp-c-danger-soft);--vp-code-copy-code-border-color: var(--vp-c-divider);--vp-code-copy-code-bg: var(--vp-c-bg-soft);--vp-code-copy-code-hover-border-color: var(--vp-c-divider);--vp-code-copy-code-hover-bg: var(--vp-c-bg);--vp-code-copy-code-active-text: var(--vp-c-text-2);--vp-code-copy-copied-text-content: "Copied";--vp-code-tab-divider: var(--vp-code-block-divider-color);--vp-code-tab-text-color: var(--vp-c-text-2);--vp-code-tab-bg: var(--vp-code-block-bg);--vp-code-tab-hover-text-color: var(--vp-c-text-1);--vp-code-tab-active-text-color: var(--vp-c-text-1);--vp-code-tab-active-bar-color: var(--vp-c-brand-1)}:lang(es),:lang(pt){--vp-code-copy-copied-text-content: "Copiado"}:lang(fa){--vp-code-copy-copied-text-content: "کپی شد"}:lang(ko){--vp-code-copy-copied-text-content: "복사됨"}:lang(ru){--vp-code-copy-copied-text-content: "Скопировано"}:lang(zh){--vp-code-copy-copied-text-content: "已复制"}:root{--vp-button-brand-border: transparent;--vp-button-brand-text: var(--vp-c-white);--vp-button-brand-bg: var(--vp-c-brand-3);--vp-button-brand-hover-border: transparent;--vp-button-brand-hover-text: var(--vp-c-white);--vp-button-brand-hover-bg: var(--vp-c-brand-2);--vp-button-brand-active-border: transparent;--vp-button-brand-active-text: var(--vp-c-white);--vp-button-brand-active-bg: var(--vp-c-brand-1);--vp-button-alt-border: transparent;--vp-button-alt-text: var(--vp-c-text-1);--vp-button-alt-bg: var(--vp-c-default-3);--vp-button-alt-hover-border: transparent;--vp-button-alt-hover-text: var(--vp-c-text-1);--vp-button-alt-hover-bg: var(--vp-c-default-2);--vp-button-alt-active-border: transparent;--vp-button-alt-active-text: var(--vp-c-text-1);--vp-button-alt-active-bg: var(--vp-c-default-1);--vp-button-sponsor-border: var(--vp-c-text-2);--vp-button-sponsor-text: var(--vp-c-text-2);--vp-button-sponsor-bg: transparent;--vp-button-sponsor-hover-border: var(--vp-c-sponsor);--vp-button-sponsor-hover-text: var(--vp-c-sponsor);--vp-button-sponsor-hover-bg: transparent;--vp-button-sponsor-active-border: var(--vp-c-sponsor);--vp-button-sponsor-active-text: var(--vp-c-sponsor);--vp-button-sponsor-active-bg: transparent}:root{--vp-custom-block-font-size: 14px;--vp-custom-block-code-font-size: 13px;--vp-custom-block-info-border: transparent;--vp-custom-block-info-text: var(--vp-c-text-1);--vp-custom-block-info-bg: var(--vp-c-default-soft);--vp-custom-block-info-code-bg: var(--vp-c-default-soft);--vp-custom-block-note-border: transparent;--vp-custom-block-note-text: var(--vp-c-text-1);--vp-custom-block-note-bg: var(--vp-c-default-soft);--vp-custom-block-note-code-bg: var(--vp-c-default-soft);--vp-custom-block-tip-border: transparent;--vp-custom-block-tip-text: var(--vp-c-text-1);--vp-custom-block-tip-bg: var(--vp-c-tip-soft);--vp-custom-block-tip-code-bg: var(--vp-c-tip-soft);--vp-custom-block-important-border: transparent;--vp-custom-block-important-text: var(--vp-c-text-1);--vp-custom-block-important-bg: var(--vp-c-important-soft);--vp-custom-block-important-code-bg: var(--vp-c-important-soft);--vp-custom-block-warning-border: transparent;--vp-custom-block-warning-text: var(--vp-c-text-1);--vp-custom-block-warning-bg: 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l}from"./chunks/framework.BI0fNMXE.js";const d=JSON.parse('{"title":"Getting Started","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/1_GettingStarted.md","filePath":"tutorials/1_GettingStarted.md","lastUpdated":null}'),e={name:"tutorials/1_GettingStarted.md"};function i(t,s,r,c,o,h){return l(),n("div",null,s[0]||(s[0]=[p(`

Getting Started ​

Prerequisites

Here we assume that you are familiar with Lux.jl. If not please take a look at the Lux.jl tutoials.

Boltz.jl is just like Lux.jl but comes with more "batteries included". Let's start by defining an MLP model.

julia
using Lux, Boltz, Random

Multi-Layer Perceptron ​

If we were to do this in Lux.jl we would write the following:

julia
model = Chain(
+    Dense(784, 256, relu),
+    Dense(256, 10)
+)
Chain(
+    layer_1 = Dense(784 => 256, relu),  # 200_960 parameters
+    layer_2 = Dense(256 => 10),         # 2_570 parameters
+)         # Total: 203_530 parameters,
+          #        plus 0 states.

But in Boltz.jl we can do this:

julia
model = Layers.MLP(784, (256, 10), relu)
MLP(
+    chain = Chain(
+        block1 = DenseNormActDropoutBlock(
+            block = Chain(
+                dense = Dense(784 => 256, relu),  # 200_960 parameters
+            ),
+        ),
+        block2 = DenseNormActDropoutBlock(
+            block = Chain(
+                dense = Dense(256 => 10),  # 2_570 parameters
+            ),
+        ),
+    ),
+)         # Total: 203_530 parameters,
+          #        plus 0 states.

The MLP function is just a convenience wrapper around Lux.Chain that constructs a multi-layer perceptron with the given number of layers and activation function.

How about VGG? ​

Let's take a look at the Vision module. We can construct a VGG model with the following code:

julia
Vision.VGG(13)
VGG(
+    layer = Chain(
+        feature_extractor = VGGFeatureExtractor(
+            model = Chain(
+                layer_1 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 3 => 64, relu, pad=1),  # 1_792 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 64, relu, pad=1),  # 36_928 parameters
+                        ),
+                    ),
+                ),
+                layer_2 = MaxPool((2, 2)),
+                layer_3 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 128, relu, pad=1),  # 73_856 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 128, relu, pad=1),  # 147_584 parameters
+                        ),
+                    ),
+                ),
+                layer_4 = MaxPool((2, 2)),
+                layer_5 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 256, relu, pad=1),  # 295_168 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 256, relu, pad=1),  # 590_080 parameters
+                        ),
+                    ),
+                ),
+                layer_6 = MaxPool((2, 2)),
+                layer_7 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 512, relu, pad=1),  # 1_180_160 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_8 = MaxPool((2, 2)),
+                layer_9 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_10 = MaxPool((2, 2)),
+            ),
+        ),
+        classifier = VGGClassifier(
+            model = Chain(
+                layer_1 = Lux.FlattenLayer{Nothing}(nothing),
+                layer_2 = Dense(25088 => 4096, relu),  # 102_764_544 parameters
+                layer_3 = Dropout(0.5),
+                layer_4 = Dense(4096 => 4096, relu),  # 16_781_312 parameters
+                layer_5 = Dropout(0.5),
+                layer_6 = Dense(4096 => 1000),  # 4_097_000 parameters
+            ),
+        ),
+    ),
+)         # Total: 133_047_848 parameters,
+          #        plus 4 states.

We can also load pretrained ImageNet weights using

Load JLD2

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

julia
using JLD2
+
+Vision.VGG(13; pretrained=true)
VGG(
+    layer = Chain(
+        feature_extractor = VGGFeatureExtractor(
+            model = Chain(
+                layer_1 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 3 => 64, relu, pad=1),  # 1_792 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 64, relu, pad=1),  # 36_928 parameters
+                        ),
+                    ),
+                ),
+                layer_2 = MaxPool((2, 2)),
+                layer_3 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 128, relu, pad=1),  # 73_856 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 128, relu, pad=1),  # 147_584 parameters
+                        ),
+                    ),
+                ),
+                layer_4 = MaxPool((2, 2)),
+                layer_5 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 256, relu, pad=1),  # 295_168 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 256, relu, pad=1),  # 590_080 parameters
+                        ),
+                    ),
+                ),
+                layer_6 = MaxPool((2, 2)),
+                layer_7 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 512, relu, pad=1),  # 1_180_160 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_8 = MaxPool((2, 2)),
+                layer_9 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_10 = MaxPool((2, 2)),
+            ),
+        ),
+        classifier = VGGClassifier(
+            model = Chain(
+                layer_1 = Lux.FlattenLayer{Nothing}(nothing),
+                layer_2 = Dense(25088 => 4096, relu),  # 102_764_544 parameters
+                layer_3 = Dropout(0.5),
+                layer_4 = Dense(4096 => 4096, relu),  # 16_781_312 parameters
+                layer_5 = Dropout(0.5),
+                layer_6 = Dense(4096 => 1000),  # 4_097_000 parameters
+            ),
+        ),
+    ),
+)         # Total: 133_047_848 parameters,
+          #        plus 4 states.

Loading Models from Metalhead (Flux.jl) ​

We can load models from Metalhead (Flux.jl), just remember to load Metalhead before.

julia
using Metalhead
+
+Vision.ResNet(18)
MetalheadWrapperLayer(
+    layer = Chain(
+        layer_1 = Chain(
+            layer_1 = Chain(
+                layer_1 = Conv((7, 7), 3 => 64, pad=3, stride=2, use_bias=false),  # 9_408 parameters
+                layer_2 = BatchNorm(64, relu, affine=true, track_stats=true),  # 128 parameters, plus 129
+                layer_3 = MaxPool((3, 3), pad=1, stride=2),
+            ),
+            layer_2 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_2 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_5 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_2 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_5 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                    ),
+                ),
+            ),
+            layer_3 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 64 => 128, stride=2, use_bias=false),  # 8_192 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 128, pad=1, stride=2, use_bias=false),  # 73_728 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_5 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_5 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                ),
+            ),
+            layer_4 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 128 => 256, stride=2, use_bias=false),  # 32_768 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 128 => 256, pad=1, stride=2, use_bias=false),  # 294_912 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_5 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_5 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                ),
+            ),
+            layer_5 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 256 => 512, stride=2, use_bias=false),  # 131_072 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 256 => 512, pad=1, stride=2, use_bias=false),  # 1_179_648 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_5 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_5 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                ),
+            ),
+        ),
+        layer_2 = Chain(
+            layer_1 = AdaptiveMeanPool((1, 1)),
+            layer_2 = WrappedFunction(flatten),
+            layer_3 = Dense(512 => 1000),  # 513_000 parameters
+        ),
+    ),
+)         # Total: 11_689_512 parameters,
+          #        plus 9_620 states.

Appendix ​

julia
using InteractiveUtils
+InteractiveUtils.versioninfo()
+
+if @isdefined(MLDataDevices)
+    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
+        println()
+        CUDA.versioninfo()
+    end
+
+    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
+        println()
+        AMDGPU.versioninfo()
+    end
+end
Julia Version 1.11.3
+Commit d63adeda50d (2025-01-21 19:42 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  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.

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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

',3)),s("mjx-container",p,[(t(),A("svg",Q,a[0]||(a[0]=[n('',1)]))),a[1]||(a[1]=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("msup",null,[s("mi",null,"x"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-alternate":"1"},"′"),s("mi",{"data-mjx-alternate":"1"},"′")])]),s("mo",null,"="),s("msup",null,[s("mi",null,"u"),s("mn",null,"3")]),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),s("p",null,[a[4]||(a[4]=i("where we want to optimize our controller ")),s("mjx-container",r,[(t(),A("svg",T,a[2]||(a[2]=[n('',1)]))),a[3]||(a[3]=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("mi",null,"u"),s("mo",{stretchy:"false"},"("),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),a[5]||(a[5]=i(" such that the following is minimized:"))]),s("mjx-container",h,[(t(),A("svg",d,a[6]||(a[6]=[n('',1)]))),a[7]||(a[7]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")),s("mjx-container",o,[(t(),A("svg",k,a[8]||(a[8]=[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":"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)]))),a[9]||(a[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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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:"))]),a[38]||(a[38]=n(`

Package Imports ​

julia
using Lux, Boltz, ComponentArrays, OrdinaryDiffEqVerner, Optimization, OptimizationOptimJL,
+      OptimizationOptimisers, SciMLSensitivity, Statistics, Printf, Random
+using DynamicExpressions, SymbolicRegression, MLJ, SymbolicUtils, Latexify
+using CairoMakie
Precompiling ComponentArrays...
+    840.2 ms  ✓ ComponentArrays
+  1 dependency successfully precompiled in 1 seconds. 45 already precompiled.
+Precompiling MLDataDevicesComponentArraysExt...
+    535.7 ms  ✓ MLDataDevices → MLDataDevicesComponentArraysExt
+  1 dependency successfully precompiled in 1 seconds. 48 already precompiled.
+Precompiling LuxComponentArraysExt...
+    651.1 ms  ✓ ComponentArrays → ComponentArraysOptimisersExt
+   1240.5 ms  ✓ Lux → LuxComponentArraysExt
+   2063.7 ms  ✓ ComponentArrays → ComponentArraysKernelAbstractionsExt
+  3 dependencies successfully precompiled in 2 seconds. 111 already precompiled.
+Precompiling OrdinaryDiffEqVerner...
+    458.0 ms  ✓ SimpleUnPack
+    501.0 ms  ✓ UnPack
+    516.1 ms  ✓ CommonSolve
+    577.1 ms  ✓ ExprTools
+    435.9 ms  ✓ FastPower
+    524.3 ms  ✓ EnumX
+    515.5 ms  ✓ MuladdMacro
+   1090.9 ms  ✓ FunctionWrappers
+    580.5 ms  ✓ SciMLStructures
+   1060.2 ms  ✓ TruncatedStacktraces
+   1086.0 ms  ✓ PreallocationTools
+    878.7 ms  ✓ Parameters
+   1398.2 ms  ✓ FastBroadcast
+   2640.0 ms  ✓ RecipesBase
+    682.5 ms  ✓ RuntimeGeneratedFunctions
+   3076.8 ms  ✓ SciMLOperators
+    646.3 ms  ✓ FunctionWrappersWrappers
+    932.7 ms  ✓ FastPower → FastPowerForwardDiffExt
+    731.8 ms  ✓ SciMLOperators → SciMLOperatorsStaticArraysCoreExt
+   1583.0 ms  ✓ SymbolicIndexingInterface
+   1902.9 ms  ✓ RecursiveArrayTools
+    963.4 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsForwardDiffExt
+    974.4 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsFastBroadcastExt
+   9110.3 ms  ✓ Expronicon
+  10995.3 ms  ✓ SciMLBase
+   5467.3 ms  ✓ DiffEqBase
+   4219.0 ms  ✓ OrdinaryDiffEqCore
+   1267.4 ms  ✓ OrdinaryDiffEqCore → OrdinaryDiffEqCoreEnzymeCoreExt
+  34662.9 ms  ✓ OrdinaryDiffEqVerner
+  29 dependencies successfully precompiled in 67 seconds. 96 already precompiled.
+Precompiling MLDataDevicesRecursiveArrayToolsExt...
+    499.4 ms  ✓ MLDataDevices → MLDataDevicesRecursiveArrayToolsExt
+  1 dependency successfully precompiled in 1 seconds. 47 already precompiled.
+Precompiling ComponentArraysRecursiveArrayToolsExt...
+    577.7 ms  ✓ ComponentArrays → ComponentArraysRecursiveArrayToolsExt
+  1 dependency successfully precompiled in 1 seconds. 69 already precompiled.
+Precompiling ComponentArraysSciMLBaseExt...
+    850.8 ms  ✓ SciMLBase → SciMLBaseChainRulesCoreExt
+    968.9 ms  ✓ ComponentArrays → ComponentArraysSciMLBaseExt
+  2 dependencies successfully precompiled in 1 seconds. 97 already precompiled.
+Precompiling DiffEqBaseChainRulesCoreExt...
+   1291.5 ms  ✓ DiffEqBase → DiffEqBaseChainRulesCoreExt
+  1 dependency successfully precompiled in 2 seconds. 125 already precompiled.
+Precompiling Optimization...
+    638.6 ms  ✓ LeftChildRightSiblingTrees
+    755.4 ms  ✓ LoggingExtras
+   1363.6 ms  ✓ DifferentiationInterface
+   1430.7 ms  ✓ ProgressMeter
+   1569.4 ms  ✓ PDMats
+   1215.5 ms  ✓ SciMLOperators → SciMLOperatorsSparseArraysExt
+   1131.2 ms  ✓ L_BFGS_B_jll
+   1503.5 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsSparseArraysExt
+   1229.8 ms  ✓ TerminalLoggers
+   1970.1 ms  ✓ SparseMatrixColorings
+    919.8 ms  ✓ DifferentiationInterface → DifferentiationInterfaceSparseArraysExt
+    804.0 ms  ✓ ConsoleProgressMonitor
+   1000.2 ms  ✓ FillArrays → FillArraysPDMatsExt
+    811.5 ms  ✓ LBFGSB
+   1096.5 ms  ✓ DifferentiationInterface → DifferentiationInterfaceSparseMatrixColoringsExt
+   4862.0 ms  ✓ SparseConnectivityTracer
+   1904.0 ms  ✓ OptimizationBase
+   1765.7 ms  ✓ Optimization
+  18 dependencies successfully precompiled in 9 seconds. 86 already precompiled.
+Precompiling DiffEqBaseSparseArraysExt...
+   1337.3 ms  ✓ DiffEqBase → DiffEqBaseSparseArraysExt
+  1 dependency successfully precompiled in 2 seconds. 125 already precompiled.
+Precompiling DifferentiationInterfaceChainRulesCoreExt...
+    366.9 ms  ✓ DifferentiationInterface → DifferentiationInterfaceChainRulesCoreExt
+  1 dependency successfully precompiled in 0 seconds. 11 already precompiled.
+Precompiling DifferentiationInterfaceStaticArraysExt...
+    513.8 ms  ✓ DifferentiationInterface → DifferentiationInterfaceStaticArraysExt
+  1 dependency successfully precompiled in 1 seconds. 10 already precompiled.
+Precompiling DifferentiationInterfaceForwardDiffExt...
+    700.6 ms  ✓ DifferentiationInterface → DifferentiationInterfaceForwardDiffExt
+  1 dependency successfully precompiled in 1 seconds. 28 already precompiled.
+Precompiling SparseConnectivityTracerSpecialFunctionsExt...
+   1040.3 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerLogExpFunctionsExt
+   1406.7 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerSpecialFunctionsExt
+  2 dependencies successfully precompiled in 2 seconds. 26 already precompiled.
+Precompiling SparseConnectivityTracerNNlibExt...
+   1415.1 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerNNlibExt
+  1 dependency successfully precompiled in 2 seconds. 46 already precompiled.
+Precompiling SparseConnectivityTracerNaNMathExt...
+   1102.4 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerNaNMathExt
+  1 dependency successfully precompiled in 1 seconds. 18 already precompiled.
+Precompiling OptimizationForwardDiffExt...
+    556.5 ms  ✓ OptimizationBase → OptimizationForwardDiffExt
+  1 dependency successfully precompiled in 1 seconds. 110 already precompiled.
+Precompiling OptimizationMLDataDevicesExt...
+   1197.0 ms  ✓ OptimizationBase → OptimizationMLDataDevicesExt
+  1 dependency successfully precompiled in 1 seconds. 97 already precompiled.
+Precompiling OptimizationOptimJL...
+    394.6 ms  ✓ PositiveFactorizations
+    558.8 ms  ✓ FiniteDiff
+    531.1 ms  ✓ OptimizationBase → OptimizationFiniteDiffExt
+    574.3 ms  ✓ DifferentiationInterface → DifferentiationInterfaceFiniteDiffExt
+    747.0 ms  ✓ FiniteDiff → FiniteDiffSparseArraysExt
+   1113.7 ms  ✓ NLSolversBase
+   1605.8 ms  ✓ LineSearches
+   2870.3 ms  ✓ Optim
+  15169.9 ms  ✓ OptimizationOptimJL
+  9 dependencies successfully precompiled in 22 seconds. 131 already precompiled.
+Precompiling FiniteDiffStaticArraysExt...
+    511.2 ms  ✓ FiniteDiff → FiniteDiffStaticArraysExt
+  1 dependency successfully precompiled in 1 seconds. 21 already precompiled.
+Precompiling OptimizationOptimisers...
+   1669.6 ms  ✓ OptimizationOptimisers
+  1 dependency successfully precompiled in 2 seconds. 113 already precompiled.
+Precompiling SciMLSensitivity...
+    594.4 ms  ✓ PoissonRandom
+    732.5 ms  ✓ StructIO
+   1640.6 ms  ✓ OffsetArrays
+   1809.3 ms  ✓ Cassette
+   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.
+Precompiling LuxEnzymeExt...
+   5931.9 ms  ✓ Lux → LuxEnzymeExt
+  1 dependency successfully precompiled in 6 seconds. 146 already precompiled.
+Precompiling OptimizationEnzymeExt...
+  11637.8 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
+  1 dependency successfully precompiled in 2 seconds. 114 already precompiled.
+Precompiling BoltzTrackerExt...
+   2073.4 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
+  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
+  1 dependency successfully precompiled in 3 seconds. 130 already precompiled.
+Precompiling ComponentArraysZygoteExt...
+   1398.7 ms  ✓ ComponentArrays → ComponentArraysZygoteExt
+   1667.0 ms  ✓ ComponentArrays → ComponentArraysGPUArraysExt
+  2 dependencies successfully precompiled in 2 seconds. 116 already precompiled.
+Precompiling OptimizationZygoteExt...
+   1873.6 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
+  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.
+Precompiling LuxLossFunctionsExt...
+   1369.9 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.
+Precompiling OptimizationMLUtilsExt...
+   1592.6 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
+  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
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Helper Functions ​

julia
function plot_dynamics(sol, us, ts)
+    fig = Figure()
+    ax = CairoMakie.Axis(fig[1, 1]; xlabel=L"t")
+    ylims!(ax, (-6, 6))
+
+    lines!(ax, ts, sol[1, :]; label=L"u_1(t)", linewidth=3)
+    lines!(ax, ts, sol[2, :]; label=L"u_2(t)", linewidth=3)
+
+    lines!(ax, ts, vec(us); label=L"u(t)", linewidth=3)
+
+    axislegend(ax; position=:rb)
+
+    return fig
+end
plot_dynamics (generic function with 1 method)

Training a Neural Network based UDE ​

Let's setup the neural network. For the first part, we won't do any symbolic regression. We will plain and simple train a neural network to solve the optimal control problem.

julia
rng = Xoshiro(0)
+tspan = (0.0, 8.0)
+
+mlp = Chain(Dense(1 => 4, gelu), Dense(4 => 4, gelu), Dense(4 => 1))
+
+function construct_ude(mlp, solver; kwargs...)
+    return @compact(; mlp, solver, kwargs...) do x_in, ps
+        x, ts, ret_sol = x_in
+
+        function dudt(du, u, p, t)
+            u₁, u₂ = u
+            du[1] = u₂
+            du[2] = mlp([t], p)[1]^3
+            return
+        end
+
+        prob = ODEProblem{true}(dudt, x, extrema(ts), ps.mlp)
+
+        sol = solve(prob, solver; saveat=ts,
+            sensealg=QuadratureAdjoint(; autojacvec=ReverseDiffVJP(true)), kwargs...)
+
+        us = mlp(reshape(ts, 1, :), ps.mlp)
+        ret_sol === Val(true) && @return sol, us
+        @return Array(sol), us
+    end
+end
+
+ude = construct_ude(mlp, Vern9(); abstol=1e-10, reltol=1e-10);

Here we are going to tuse the same configuration for testing, but this is to show that we can setup them up with different ode solve configurations

julia
ude_test = construct_ude(mlp, Vern9(); abstol=1e-10, reltol=1e-10);
+
+function train_model_1(ude, rng, ts_)
+    ps, st = Lux.setup(rng, ude)
+    ps = ComponentArray{Float64}(ps)
+    stateful_ude = StatefulLuxLayer{true}(ude, nothing, st)
+
+    ts = collect(ts_)
+
+    function loss_adjoint(θ)
+        x, us = stateful_ude(([-4.0, 0.0], ts, Val(false)), θ)
+        return mean(abs2, 4 .- x[1, :]) + 2 * mean(abs2, x[2, :]) + 0.1 * mean(abs2, us)
+    end
+
+    callback = function (state, l)
+        state.iter % 50 == 1 && @printf "Iteration: %5d\\tLoss: %10g\\n" state.iter l
+        return false
+    end
+
+    optf = OptimizationFunction((x, p) -> loss_adjoint(x), AutoZygote())
+    optprob = OptimizationProblem(optf, ps)
+    res1 = solve(optprob, Optimisers.Adam(0.001); callback, maxiters=500)
+
+    optprob = OptimizationProblem(optf, res1.u)
+    res2 = solve(optprob, LBFGS(); callback, maxiters=100)
+
+    return StatefulLuxLayer{true}(ude, res2.u, st)
+end
+
+trained_ude = train_model_1(ude, rng, 0.0:0.01:8.0)
┌ Warning: Lux.apply(m::AbstractLuxLayer, x::AbstractArray{<:ReverseDiff.TrackedReal}, ps, st) input was corrected to Lux.apply(m::AbstractLuxLayer, x::ReverseDiff.TrackedArray}, ps, st).
+│ 
+│ 1. If this was not the desired behavior overload the dispatch on \`m\`.
+│ 
+│ 2. This might have performance implications. Check which layer was causing this problem using \`Lux.Experimental.@debug_mode\`.
+└ @ LuxCoreArrayInterfaceReverseDiffExt ~/.julia/packages/LuxCore/8mVob/ext/LuxCoreArrayInterfaceReverseDiffExt.jl:10
+Iteration:     1	Loss:    40.5618
+Iteration:    51	Loss:    29.4147
+Iteration:   101	Loss:    28.2559
+Iteration:   151	Loss:     27.217
+Iteration:   201	Loss:    26.1657
+Iteration:   251	Loss:    25.1631
+Iteration:   301	Loss:    24.2914
+Iteration:   351	Loss:    23.5965
+Iteration:   401	Loss:    23.0763
+Iteration:   451	Loss:    22.6983
+Iteration:     1	Loss:     22.245
+Iteration:    51	Loss:    12.0107
julia
sol, us = ude_test(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), trained_ude.ps, trained_ude.st)[1];
+plot_dynamics(sol, us, 0.0:0.01:8.0)

Now that the system is in a better behaved part of parameter space, we return to the original loss function to finish the optimization:

julia
function train_model_2(stateful_ude::StatefulLuxLayer, ts_)
+    ts = collect(ts_)
+
+    function loss_adjoint(θ)
+        x, us = stateful_ude(([-4.0, 0.0], ts, Val(false)), θ)
+        return mean(abs2, 4 .- x[1, :]) .+ 2 * mean(abs2, x[2, :]) .+ mean(abs2, us)
+    end
+
+    callback = function (state, l)
+        state.iter % 10 == 1 && @printf "Iteration: %5d\\tLoss: %10g\\n" state.iter l
+        return false
+    end
+
+    optf = OptimizationFunction((x, p) -> loss_adjoint(x), AutoZygote())
+    optprob = OptimizationProblem(optf, stateful_ude.ps)
+    res2 = solve(optprob, LBFGS(); callback, maxiters=100)
+
+    return StatefulLuxLayer{true}(stateful_ude.model, res2.u, stateful_ude.st)
+end
+
+trained_ude = train_model_2(trained_ude, 0.0:0.01:8.0)
┌ Warning: Lux.apply(m::AbstractLuxLayer, x::AbstractArray{<:ReverseDiff.TrackedReal}, ps, st) input was corrected to Lux.apply(m::AbstractLuxLayer, x::ReverseDiff.TrackedArray}, ps, st).
+│ 
+│ 1. If this was not the desired behavior overload the dispatch on \`m\`.
+│ 
+│ 2. This might have performance implications. Check which layer was causing this problem using \`Lux.Experimental.@debug_mode\`.
+└ @ LuxCoreArrayInterfaceReverseDiffExt ~/.julia/packages/LuxCore/8mVob/ext/LuxCoreArrayInterfaceReverseDiffExt.jl:10
+Iteration:     1	Loss:    12.5986
+Iteration:    11	Loss:     12.559
+Iteration:    21	Loss:     12.471
+Iteration:    31	Loss:    12.4164
+Iteration:    41	Loss:     12.401
+Iteration:    51	Loss:    12.3783
+Iteration:    61	Loss:    12.3635
+Iteration:    71	Loss:    12.3562
+Iteration:    81	Loss:    12.3519
+Iteration:    91	Loss:    12.3471
julia
sol, us = ude_test(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), trained_ude.ps, trained_ude.st)[1];
+plot_dynamics(sol, us, 0.0:0.01:8.0)

Symbolic Regression ​

Ok so now we have a trained neural network that solves the optimal control problem. But can we replace Dense(4 => 4, gelu) with a symbolic expression? Let's try!

Data Generation for Symbolic Regression ​

First, we need to generate data for the symbolic regression.

julia
ts = reshape(collect(0.0:0.1:8.0), 1, :)
+
+X_train = mlp[1](ts, trained_ude.ps.mlp.layer_1, trained_ude.st.mlp.layer_1)[1]
4×81 Matrix{Float64}:
+ -0.0397511  -0.0317854  -0.0250462  -0.0194458  -0.014873   -0.0112035   -0.00830924   -0.00606573   -0.00435673  -0.00307771   -0.00213748   -0.00145881  -0.000977957  -0.00064367  -0.00041574  -0.000263381  -0.000163581  -9.95524e-5  -5.93359e-5   -3.46184e-5   -1.97604e-5   -1.10295e-5   -6.01676e-6   -3.20614e-6   -1.66797e-6   -8.46734e-7   -4.19207e-7   -2.02301e-7   -9.51099e-8   -4.35387e-8   -1.9396e-8    -8.40435e-9   -3.54008e-9   -1.44879e-9   -5.75761e-10  -2.2207e-10   -8.30827e-11  -3.01347e-11  -1.05906e-11  -3.60445e-12  -1.18735e-12  -3.78357e-13  -1.16566e-13  -3.47018e-14  -9.977e-15  -2.76872e-15  -7.41222e-16  -1.91325e-16  -4.75889e-17  -1.14002e-17  -2.62879e-18  -5.83165e-19  -1.24389e-19  -2.54967e-20  -5.01952e-21  -9.48579e-22  -1.7198e-22   -2.98976e-23  -4.98088e-24  -7.94783e-25  -1.21401e-25  -1.77414e-26  -2.47914e-27  -3.31074e-28  -4.22297e-29  -5.14207e-30  -5.97371e-31  -6.61753e-32  -6.98637e-33  -7.0254e-34   -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0
+  0.111436    0.0419072  -0.017267   -0.0660352  -0.104617   -0.13349     -0.153362     -0.165131     -0.169837    -0.168603     -0.162587     -0.152923    -0.140682     -0.126833    -0.112217    -0.0975332    -0.0833352    -0.0700333   -0.057906     -0.0471162    -0.0377292    -0.0297327    -0.0230565    -0.0175905    -0.0132002    -0.00974017   -0.00706458   -0.00503462   -0.00352389   -0.00242131   -0.00163244   -0.00107935   -0.000699517  -0.000444123  -0.000276081  -0.000167939  -9.99068e-5   -5.8092e-5    -3.29957e-5   -1.82961e-5   -9.89833e-6   -5.2216e-6    -2.68423e-6   -1.34384e-6   -6.5482e-7  -3.10366e-7   -1.43001e-7   -6.40098e-8   -2.78182e-8   -1.17305e-8   -4.7967e-9    -1.90078e-9   -7.29484e-10  -2.70971e-10  -9.73601e-11  -3.38157e-11  -1.13465e-11  -3.67571e-12  -1.14889e-12  -3.46263e-13  -1.00565e-13  -2.81272e-14  -7.57134e-15  -1.96024e-15  -4.87825e-16  -1.16617e-16  -2.67624e-17  -5.89225e-18  -1.24381e-18  -2.51577e-19  -4.87251e-20  -9.03078e-21  -1.60071e-21  -2.71169e-22  -4.38764e-23  -6.77655e-24  -9.98384e-25  -1.40224e-25  -1.87631e-26  -2.39039e-27  -2.89761e-28
+ -0.110793   -0.0822512  -0.057209   -0.0373554  -0.0229031  -0.0131703   -0.00708815   -0.00355998   -0.00166277  -0.000719452  -0.000287171  -0.00010528  -3.52903e-5   -1.07663e-5  -2.97538e-6  -7.4137e-7    -1.65756e-7   -3.30953e-8  -5.87257e-9   -9.21623e-10  -1.273e-10    -1.54007e-11  -1.62392e-12  -1.48515e-13  -1.17228e-14  -7.9471e-16   -4.60432e-17  -2.26861e-18  -9.45911e-20  -3.32116e-21  -9.77079e-23  -2.39675e-24  -4.87773e-26  -8.19525e-28  -1.13111e-29  -1.2761e-31   -1.171e-33    -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0        -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0
+ -0.076181   -0.0468178  -0.026027   -0.0130715  -0.0059069  -0.00238742  -0.000856747  -0.000270747  -7.46883e-5  -1.78221e-5   -3.64438e-6   -6.32607e-7  -9.23267e-8   -1.12204e-8  -1.12451e-9  -9.20332e-11  -6.09117e-12  -3.2282e-13  -1.35657e-14  -4.47562e-16  -1.14788e-17  -2.26603e-19  -3.40918e-21  -3.87018e-23  -3.28239e-25  -2.05921e-27  -9.46098e-30  -3.15184e-32  -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0        -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0

This is the training input data. Now we generate the targets

julia
Y_train = mlp[2](X_train, trained_ude.ps.mlp.layer_2, trained_ude.st.mlp.layer_2)[1]
4×81 Matrix{Float64}:
+  0.0516886   0.045542   0.0400219   0.0359023   0.0332579   0.0317798   0.0310197   0.0305495   0.0300455   0.0293137   0.0282779   0.0269476   0.0253838   0.0236698   0.0218909   0.0201231   0.0184276   0.0168498   0.0154194   0.0141528   0.0130554   0.0121237   0.011348    0.0107144   0.0102065   0.00980671   0.00949794   0.00926388   0.00908979   0.0089628   0.00887198   0.00880831   0.0087646   0.00873521   0.00871587   0.00870343   0.0086956   0.00869079   0.0086879   0.00868621   0.00868524   0.00868471   0.00868441   0.00868426   0.00868418   0.00868414   0.00868412   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411
+ -0.166848   -0.15062   -0.130688   -0.112766   -0.0986722  -0.0884998  -0.081764   -0.0778875  -0.0763561  -0.0767414  -0.0786804  -0.0818489  -0.0859417  -0.0906647  -0.0957383  -0.100907   -0.105953   -0.110702   -0.115033   -0.118875   -0.122199   -0.125009   -0.127337   -0.129228   -0.130735   -0.131915    -0.132823    -0.133509    -0.134017    -0.134388   -0.134652    -0.134837    -0.134964   -0.135049    -0.135105    -0.135142    -0.135164   -0.135178    -0.135187   -0.135191    -0.135194    -0.135196    -0.135197    -0.135197    -0.135197    -0.135197    -0.135197    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198
+  1.15297     1.25565    1.33772     1.40111     1.44858     1.4827      1.50547     1.51843     1.52286     1.51998     1.5111      1.49754     1.48062     1.4616      1.44157     1.42147     1.40205     1.38386     1.36727     1.35251     1.33967     1.32874     1.31961     1.31213     1.30613     1.3014       1.29774      1.29496      1.2929       1.29139     1.29031      1.28956      1.28904     1.28869      1.28846      1.28831      1.28822     1.28816      1.28813     1.28811      1.28809      1.28809      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808
+  0.884722    0.264828  -0.0595685  -0.162055   -0.165213   -0.142734   -0.121854   -0.10885    -0.103638   -0.10493    -0.111445   -0.121894   -0.134731   -0.148027   -0.159607   -0.167429   -0.170034   -0.166869   -0.158333   -0.145568   -0.130106   -0.113513   -0.0971338  -0.0819568  -0.0685907  -0.0573086   -0.0481281   -0.040896    -0.0353637   -0.0312451  -0.0282564   -0.02614     -0.0246766  -0.0236881   -0.0230357   -0.0226149   -0.0223499  -0.0221869   -0.022089   -0.0220317   -0.0219989   -0.0219807   -0.0219708   -0.0219655   -0.0219629   -0.0219615   -0.0219609   -0.0219605   -0.0219604   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603

Fitting the Symbolic Expression ​

We will follow the example from SymbolicRegression.jl docs to fit the symbolic expression.

julia
srmodel = MultitargetSRRegressor(;
+    binary_operators=[+, -, *, /], niterations=100, save_to_file=false);

One important note here is to transpose the data because that is how MLJ expects the data to be structured (this is in contrast to how Lux or SymbolicRegression expects the data)

julia
mach = machine(srmodel, X_train', Y_train')
+fit!(mach; verbosity=0)
+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),
+    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_2 = WrappedFunction(stack1),
+        ),
+    ),
+    layer_3 = Dense(4 => 1),            # 5 parameters
+)         # Total: 34 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,
+        layer_2=Lux.initialparameters(rng, hybrid_mlp[2]),
+        layer_3=trained_ude.ps.mlp.layer_3))
+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
+InteractiveUtils.versioninfo()
+
+if @isdefined(MLDataDevices)
+    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
+        println()
+        CUDA.versioninfo()
+    end
+
+    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
+        println()
+        AMDGPU.versioninfo()
+    end
+end
Julia Version 1.11.3
+Commit d63adeda50d (2025-01-21 19:42 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  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.

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a/previews/PR98/hashmap.json b/previews/PR98/hashmap.json new file mode 100644 index 0000000..5da7ab6 --- /dev/null +++ b/previews/PR98/hashmap.json @@ -0,0 +1 @@ +{"api_basis.md":"D9oO8cm0","api_index.md":"Huv0Y-72","api_layers.md":"DmVxh5An","api_private.md":"BJWD8BVj","api_vision.md":"B5rI8T8X","index.md":"ac-RXHi6","tutorials_1_gettingstarted.md":"I5QDH3_W","tutorials_2_symbolicoptimalcontrol.md":"BLCy63T9"} diff --git a/previews/PR98/index.html b/previews/PR98/index.html new file mode 100644 index 0000000..93d3c90 --- /dev/null +++ b/previews/PR98/index.html @@ -0,0 +1,40 @@ + + + + + + Boltz.jl Docs + + + + + + + + + + + + + + + + + + + + + +
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Boltz.jl ⚡ DocsPre-built Deep Learning Models in Julia

Accelerate ⚡ your ML research using pre-built Deep Learning Models with Lux

Lux.jl Docs
Tutorials 📚
Vision Models 👀
Layers API 🧩
View on GitHub
Lux.jl
🔥

Powered by Lux.jl

Boltz.jl is built on top of Lux.jl, a pure Julia Deep Learning Framework designed for Scientific Machine Learning.

🧩

Pre-built Models

Boltz.jl provides pre-built models for common deep learning tasks, such as image classification.

🧑‍🔬

SciML Primitives

Common deep learning primitives needed for scientific machine learning.

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
+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")
+ + + + \ No newline at end of file diff --git a/previews/PR98/lux-logo-dark.svg b/previews/PR98/lux-logo-dark.svg new file mode 100644 index 0000000..ae8cbe3 --- /dev/null +++ b/previews/PR98/lux-logo-dark.svg @@ -0,0 +1,38 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR98/lux-logo.svg b/previews/PR98/lux-logo.svg new file mode 100644 index 0000000..0ba3c90 --- /dev/null +++ b/previews/PR98/lux-logo.svg @@ -0,0 +1,38 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR98/siteinfo.js b/previews/PR98/siteinfo.js new file mode 100644 index 0000000..79840ec --- /dev/null +++ b/previews/PR98/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR98"; diff --git a/previews/PR98/tutorials/1_GettingStarted.html b/previews/PR98/tutorials/1_GettingStarted.html new file mode 100644 index 0000000..9bad53e --- /dev/null +++ b/previews/PR98/tutorials/1_GettingStarted.html @@ -0,0 +1,351 @@ + + + + + + Getting Started | Boltz.jl Docs + + + + + + + + + + + + + + + + + + + + + +
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Getting Started ​

Prerequisites

Here we assume that you are familiar with Lux.jl. If not please take a look at the Lux.jl tutoials.

Boltz.jl is just like Lux.jl but comes with more "batteries included". Let's start by defining an MLP model.

julia
using Lux, Boltz, Random

Multi-Layer Perceptron ​

If we were to do this in Lux.jl we would write the following:

julia
model = Chain(
+    Dense(784, 256, relu),
+    Dense(256, 10)
+)
Chain(
+    layer_1 = Dense(784 => 256, relu),  # 200_960 parameters
+    layer_2 = Dense(256 => 10),         # 2_570 parameters
+)         # Total: 203_530 parameters,
+          #        plus 0 states.

But in Boltz.jl we can do this:

julia
model = Layers.MLP(784, (256, 10), relu)
MLP(
+    chain = Chain(
+        block1 = DenseNormActDropoutBlock(
+            block = Chain(
+                dense = Dense(784 => 256, relu),  # 200_960 parameters
+            ),
+        ),
+        block2 = DenseNormActDropoutBlock(
+            block = Chain(
+                dense = Dense(256 => 10),  # 2_570 parameters
+            ),
+        ),
+    ),
+)         # Total: 203_530 parameters,
+          #        plus 0 states.

The MLP function is just a convenience wrapper around Lux.Chain that constructs a multi-layer perceptron with the given number of layers and activation function.

How about VGG? ​

Let's take a look at the Vision module. We can construct a VGG model with the following code:

julia
Vision.VGG(13)
VGG(
+    layer = Chain(
+        feature_extractor = VGGFeatureExtractor(
+            model = Chain(
+                layer_1 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 3 => 64, relu, pad=1),  # 1_792 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 64, relu, pad=1),  # 36_928 parameters
+                        ),
+                    ),
+                ),
+                layer_2 = MaxPool((2, 2)),
+                layer_3 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 128, relu, pad=1),  # 73_856 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 128, relu, pad=1),  # 147_584 parameters
+                        ),
+                    ),
+                ),
+                layer_4 = MaxPool((2, 2)),
+                layer_5 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 256, relu, pad=1),  # 295_168 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 256, relu, pad=1),  # 590_080 parameters
+                        ),
+                    ),
+                ),
+                layer_6 = MaxPool((2, 2)),
+                layer_7 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 512, relu, pad=1),  # 1_180_160 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_8 = MaxPool((2, 2)),
+                layer_9 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_10 = MaxPool((2, 2)),
+            ),
+        ),
+        classifier = VGGClassifier(
+            model = Chain(
+                layer_1 = Lux.FlattenLayer{Nothing}(nothing),
+                layer_2 = Dense(25088 => 4096, relu),  # 102_764_544 parameters
+                layer_3 = Dropout(0.5),
+                layer_4 = Dense(4096 => 4096, relu),  # 16_781_312 parameters
+                layer_5 = Dropout(0.5),
+                layer_6 = Dense(4096 => 1000),  # 4_097_000 parameters
+            ),
+        ),
+    ),
+)         # Total: 133_047_848 parameters,
+          #        plus 4 states.

We can also load pretrained ImageNet weights using

Load JLD2

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

julia
using JLD2
+
+Vision.VGG(13; pretrained=true)
VGG(
+    layer = Chain(
+        feature_extractor = VGGFeatureExtractor(
+            model = Chain(
+                layer_1 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 3 => 64, relu, pad=1),  # 1_792 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 64, relu, pad=1),  # 36_928 parameters
+                        ),
+                    ),
+                ),
+                layer_2 = MaxPool((2, 2)),
+                layer_3 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 64 => 128, relu, pad=1),  # 73_856 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 128, relu, pad=1),  # 147_584 parameters
+                        ),
+                    ),
+                ),
+                layer_4 = MaxPool((2, 2)),
+                layer_5 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 128 => 256, relu, pad=1),  # 295_168 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 256, relu, pad=1),  # 590_080 parameters
+                        ),
+                    ),
+                ),
+                layer_6 = MaxPool((2, 2)),
+                layer_7 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 256 => 512, relu, pad=1),  # 1_180_160 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_8 = MaxPool((2, 2)),
+                layer_9 = ConvNormActivation(
+                    model = Chain(
+                        block1 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                        block2 = ConvNormActivationBlock(
+                            block = Conv((3, 3), 512 => 512, relu, pad=1),  # 2_359_808 parameters
+                        ),
+                    ),
+                ),
+                layer_10 = MaxPool((2, 2)),
+            ),
+        ),
+        classifier = VGGClassifier(
+            model = Chain(
+                layer_1 = Lux.FlattenLayer{Nothing}(nothing),
+                layer_2 = Dense(25088 => 4096, relu),  # 102_764_544 parameters
+                layer_3 = Dropout(0.5),
+                layer_4 = Dense(4096 => 4096, relu),  # 16_781_312 parameters
+                layer_5 = Dropout(0.5),
+                layer_6 = Dense(4096 => 1000),  # 4_097_000 parameters
+            ),
+        ),
+    ),
+)         # Total: 133_047_848 parameters,
+          #        plus 4 states.

Loading Models from Metalhead (Flux.jl) ​

We can load models from Metalhead (Flux.jl), just remember to load Metalhead before.

julia
using Metalhead
+
+Vision.ResNet(18)
MetalheadWrapperLayer(
+    layer = Chain(
+        layer_1 = Chain(
+            layer_1 = Chain(
+                layer_1 = Conv((7, 7), 3 => 64, pad=3, stride=2, use_bias=false),  # 9_408 parameters
+                layer_2 = BatchNorm(64, relu, affine=true, track_stats=true),  # 128 parameters, plus 129
+                layer_3 = MaxPool((3, 3), pad=1, stride=2),
+            ),
+            layer_2 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_2 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_5 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_2 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 64 => 64, pad=1, use_bias=false),  # 36_864 parameters
+                        layer_5 = BatchNorm(64, affine=true, track_stats=true),  # 128 parameters, plus 129
+                    ),
+                ),
+            ),
+            layer_3 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 64 => 128, stride=2, use_bias=false),  # 8_192 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 64 => 128, pad=1, stride=2, use_bias=false),  # 73_728 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_5 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_2 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 128 => 128, pad=1, use_bias=false),  # 147_456 parameters
+                        layer_5 = BatchNorm(128, affine=true, track_stats=true),  # 256 parameters, plus 257
+                    ),
+                ),
+            ),
+            layer_4 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 128 => 256, stride=2, use_bias=false),  # 32_768 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 128 => 256, pad=1, stride=2, use_bias=false),  # 294_912 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_5 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_2 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 256 => 256, pad=1, use_bias=false),  # 589_824 parameters
+                        layer_5 = BatchNorm(256, affine=true, track_stats=true),  # 512 parameters, plus 513
+                    ),
+                ),
+            ),
+            layer_5 = Chain(
+                layer_1 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Chain(
+                        layer_1 = Conv((1, 1), 256 => 512, stride=2, use_bias=false),  # 131_072 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 256 => 512, pad=1, stride=2, use_bias=false),  # 1_179_648 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_5 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                ),
+                layer_2 = Parallel(
+                    connection = addact(NNlib.relu, ...),
+                    layer_1 = Lux.NoOpLayer(),
+                    layer_2 = Chain(
+                        layer_1 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_2 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                        layer_3 = WrappedFunction(relu),
+                        layer_4 = Conv((3, 3), 512 => 512, pad=1, use_bias=false),  # 2_359_296 parameters
+                        layer_5 = BatchNorm(512, affine=true, track_stats=true),  # 1_024 parameters, plus 1_025
+                    ),
+                ),
+            ),
+        ),
+        layer_2 = Chain(
+            layer_1 = AdaptiveMeanPool((1, 1)),
+            layer_2 = WrappedFunction(flatten),
+            layer_3 = Dense(512 => 1000),  # 513_000 parameters
+        ),
+    ),
+)         # Total: 11_689_512 parameters,
+          #        plus 9_620 states.

Appendix ​

julia
using InteractiveUtils
+InteractiveUtils.versioninfo()
+
+if @isdefined(MLDataDevices)
+    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
+        println()
+        CUDA.versioninfo()
+    end
+
+    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
+        println()
+        AMDGPU.versioninfo()
+    end
+end
Julia Version 1.11.3
+Commit d63adeda50d (2025-01-21 19:42 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  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.

+ + + + \ No newline at end of file diff --git a/previews/PR98/tutorials/2_SymbolicOptimalControl.html b/previews/PR98/tutorials/2_SymbolicOptimalControl.html new file mode 100644 index 0000000..3a2bb44 --- /dev/null +++ b/previews/PR98/tutorials/2_SymbolicOptimalControl.html @@ -0,0 +1,784 @@ + + + + + + Solving Optimal Control Problems with Symbolic Universal Differential Equations | Boltz.jl Docs + + + + + + + + + + + + + + + + + + + + + +
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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

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
Precompiling ComponentArrays...
+    840.2 ms  ✓ ComponentArrays
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+Precompiling MLDataDevicesComponentArraysExt...
+    535.7 ms  ✓ MLDataDevices → MLDataDevicesComponentArraysExt
+  1 dependency successfully precompiled in 1 seconds. 48 already precompiled.
+Precompiling LuxComponentArraysExt...
+    651.1 ms  ✓ ComponentArrays → ComponentArraysOptimisersExt
+   1240.5 ms  ✓ Lux → LuxComponentArraysExt
+   2063.7 ms  ✓ ComponentArrays → ComponentArraysKernelAbstractionsExt
+  3 dependencies successfully precompiled in 2 seconds. 111 already precompiled.
+Precompiling OrdinaryDiffEqVerner...
+    458.0 ms  ✓ SimpleUnPack
+    501.0 ms  ✓ UnPack
+    516.1 ms  ✓ CommonSolve
+    577.1 ms  ✓ ExprTools
+    435.9 ms  ✓ FastPower
+    524.3 ms  ✓ EnumX
+    515.5 ms  ✓ MuladdMacro
+   1090.9 ms  ✓ FunctionWrappers
+    580.5 ms  ✓ SciMLStructures
+   1060.2 ms  ✓ TruncatedStacktraces
+   1086.0 ms  ✓ PreallocationTools
+    878.7 ms  ✓ Parameters
+   1398.2 ms  ✓ FastBroadcast
+   2640.0 ms  ✓ RecipesBase
+    682.5 ms  ✓ RuntimeGeneratedFunctions
+   3076.8 ms  ✓ SciMLOperators
+    646.3 ms  ✓ FunctionWrappersWrappers
+    932.7 ms  ✓ FastPower → FastPowerForwardDiffExt
+    731.8 ms  ✓ SciMLOperators → SciMLOperatorsStaticArraysCoreExt
+   1583.0 ms  ✓ SymbolicIndexingInterface
+   1902.9 ms  ✓ RecursiveArrayTools
+    963.4 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsForwardDiffExt
+    974.4 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsFastBroadcastExt
+   9110.3 ms  ✓ Expronicon
+  10995.3 ms  ✓ SciMLBase
+   5467.3 ms  ✓ DiffEqBase
+   4219.0 ms  ✓ OrdinaryDiffEqCore
+   1267.4 ms  ✓ OrdinaryDiffEqCore → OrdinaryDiffEqCoreEnzymeCoreExt
+  34662.9 ms  ✓ OrdinaryDiffEqVerner
+  29 dependencies successfully precompiled in 67 seconds. 96 already precompiled.
+Precompiling MLDataDevicesRecursiveArrayToolsExt...
+    499.4 ms  ✓ MLDataDevices → MLDataDevicesRecursiveArrayToolsExt
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+Precompiling ComponentArraysRecursiveArrayToolsExt...
+    577.7 ms  ✓ ComponentArrays → ComponentArraysRecursiveArrayToolsExt
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+Precompiling ComponentArraysSciMLBaseExt...
+    850.8 ms  ✓ SciMLBase → SciMLBaseChainRulesCoreExt
+    968.9 ms  ✓ ComponentArrays → ComponentArraysSciMLBaseExt
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+Precompiling DiffEqBaseChainRulesCoreExt...
+   1291.5 ms  ✓ DiffEqBase → DiffEqBaseChainRulesCoreExt
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+Precompiling Optimization...
+    638.6 ms  ✓ LeftChildRightSiblingTrees
+    755.4 ms  ✓ LoggingExtras
+   1363.6 ms  ✓ DifferentiationInterface
+   1430.7 ms  ✓ ProgressMeter
+   1569.4 ms  ✓ PDMats
+   1215.5 ms  ✓ SciMLOperators → SciMLOperatorsSparseArraysExt
+   1131.2 ms  ✓ L_BFGS_B_jll
+   1503.5 ms  ✓ RecursiveArrayTools → RecursiveArrayToolsSparseArraysExt
+   1229.8 ms  ✓ TerminalLoggers
+   1970.1 ms  ✓ SparseMatrixColorings
+    919.8 ms  ✓ DifferentiationInterface → DifferentiationInterfaceSparseArraysExt
+    804.0 ms  ✓ ConsoleProgressMonitor
+   1000.2 ms  ✓ FillArrays → FillArraysPDMatsExt
+    811.5 ms  ✓ LBFGSB
+   1096.5 ms  ✓ DifferentiationInterface → DifferentiationInterfaceSparseMatrixColoringsExt
+   4862.0 ms  ✓ SparseConnectivityTracer
+   1904.0 ms  ✓ OptimizationBase
+   1765.7 ms  ✓ Optimization
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+Precompiling DiffEqBaseSparseArraysExt...
+   1337.3 ms  ✓ DiffEqBase → DiffEqBaseSparseArraysExt
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+Precompiling DifferentiationInterfaceChainRulesCoreExt...
+    366.9 ms  ✓ DifferentiationInterface → DifferentiationInterfaceChainRulesCoreExt
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+   1406.7 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerSpecialFunctionsExt
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+Precompiling SparseConnectivityTracerNNlibExt...
+   1415.1 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerNNlibExt
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+Precompiling SparseConnectivityTracerNaNMathExt...
+   1102.4 ms  ✓ SparseConnectivityTracer → SparseConnectivityTracerNaNMathExt
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+Precompiling OptimizationForwardDiffExt...
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+   1197.0 ms  ✓ OptimizationBase → OptimizationMLDataDevicesExt
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+Precompiling OptimizationOptimJL...
+    394.6 ms  ✓ PositiveFactorizations
+    558.8 ms  ✓ FiniteDiff
+    531.1 ms  ✓ OptimizationBase → OptimizationFiniteDiffExt
+    574.3 ms  ✓ DifferentiationInterface → DifferentiationInterfaceFiniteDiffExt
+    747.0 ms  ✓ FiniteDiff → FiniteDiffSparseArraysExt
+   1113.7 ms  ✓ NLSolversBase
+   1605.8 ms  ✓ LineSearches
+   2870.3 ms  ✓ Optim
+  15169.9 ms  ✓ OptimizationOptimJL
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+   2298.9 ms  ✓ Enzyme_jll
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+   1288.4 ms  ✓ HostCPUFeatures
+  10943.2 ms  ✓ Krylov
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+   2622.4 ms  ✓ DifferentiationInterface → DifferentiationInterfaceZygoteExt
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+   6164.9 ms  ✓ FastPower → FastPowerReverseDiffExt
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+   5075.3 ms  ✓ LinearSolve → LinearSolveKernelAbstractionsExt
+ 212740.4 ms  ✓ Enzyme
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+  10286.3 ms  ✓ Enzyme → EnzymeSpecialFunctionsExt
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+  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:
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+│   Downloading artifact: IntelOpenMP
+└  
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+   2764.5 ms  ✓ LuxCore → LuxCoreArrayInterfaceReverseDiffExt
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+   1398.7 ms  ✓ ComponentArrays → ComponentArraysZygoteExt
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Helper Functions ​

julia
function plot_dynamics(sol, us, ts)
+    fig = Figure()
+    ax = CairoMakie.Axis(fig[1, 1]; xlabel=L"t")
+    ylims!(ax, (-6, 6))
+
+    lines!(ax, ts, sol[1, :]; label=L"u_1(t)", linewidth=3)
+    lines!(ax, ts, sol[2, :]; label=L"u_2(t)", linewidth=3)
+
+    lines!(ax, ts, vec(us); label=L"u(t)", linewidth=3)
+
+    axislegend(ax; position=:rb)
+
+    return fig
+end
plot_dynamics (generic function with 1 method)

Training a Neural Network based UDE ​

Let's setup the neural network. For the first part, we won't do any symbolic regression. We will plain and simple train a neural network to solve the optimal control problem.

julia
rng = Xoshiro(0)
+tspan = (0.0, 8.0)
+
+mlp = Chain(Dense(1 => 4, gelu), Dense(4 => 4, gelu), Dense(4 => 1))
+
+function construct_ude(mlp, solver; kwargs...)
+    return @compact(; mlp, solver, kwargs...) do x_in, ps
+        x, ts, ret_sol = x_in
+
+        function dudt(du, u, p, t)
+            u₁, u₂ = u
+            du[1] = u₂
+            du[2] = mlp([t], p)[1]^3
+            return
+        end
+
+        prob = ODEProblem{true}(dudt, x, extrema(ts), ps.mlp)
+
+        sol = solve(prob, solver; saveat=ts,
+            sensealg=QuadratureAdjoint(; autojacvec=ReverseDiffVJP(true)), kwargs...)
+
+        us = mlp(reshape(ts, 1, :), ps.mlp)
+        ret_sol === Val(true) && @return sol, us
+        @return Array(sol), us
+    end
+end
+
+ude = construct_ude(mlp, Vern9(); abstol=1e-10, reltol=1e-10);

Here we are going to tuse the same configuration for testing, but this is to show that we can setup them up with different ode solve configurations

julia
ude_test = construct_ude(mlp, Vern9(); abstol=1e-10, reltol=1e-10);
+
+function train_model_1(ude, rng, ts_)
+    ps, st = Lux.setup(rng, ude)
+    ps = ComponentArray{Float64}(ps)
+    stateful_ude = StatefulLuxLayer{true}(ude, nothing, st)
+
+    ts = collect(ts_)
+
+    function loss_adjoint(θ)
+        x, us = stateful_ude(([-4.0, 0.0], ts, Val(false)), θ)
+        return mean(abs2, 4 .- x[1, :]) + 2 * mean(abs2, x[2, :]) + 0.1 * mean(abs2, us)
+    end
+
+    callback = function (state, l)
+        state.iter % 50 == 1 && @printf "Iteration: %5d\tLoss: %10g\n" state.iter l
+        return false
+    end
+
+    optf = OptimizationFunction((x, p) -> loss_adjoint(x), AutoZygote())
+    optprob = OptimizationProblem(optf, ps)
+    res1 = solve(optprob, Optimisers.Adam(0.001); callback, maxiters=500)
+
+    optprob = OptimizationProblem(optf, res1.u)
+    res2 = solve(optprob, LBFGS(); callback, maxiters=100)
+
+    return StatefulLuxLayer{true}(ude, res2.u, st)
+end
+
+trained_ude = train_model_1(ude, rng, 0.0:0.01:8.0)
┌ Warning: Lux.apply(m::AbstractLuxLayer, x::AbstractArray{<:ReverseDiff.TrackedReal}, ps, st) input was corrected to Lux.apply(m::AbstractLuxLayer, x::ReverseDiff.TrackedArray}, ps, st).
+│ 
+│ 1. If this was not the desired behavior overload the dispatch on `m`.
+│ 
+│ 2. This might have performance implications. Check which layer was causing this problem using `Lux.Experimental.@debug_mode`.
+└ @ LuxCoreArrayInterfaceReverseDiffExt ~/.julia/packages/LuxCore/8mVob/ext/LuxCoreArrayInterfaceReverseDiffExt.jl:10
+Iteration:     1	Loss:    40.5618
+Iteration:    51	Loss:    29.4147
+Iteration:   101	Loss:    28.2559
+Iteration:   151	Loss:     27.217
+Iteration:   201	Loss:    26.1657
+Iteration:   251	Loss:    25.1631
+Iteration:   301	Loss:    24.2914
+Iteration:   351	Loss:    23.5965
+Iteration:   401	Loss:    23.0763
+Iteration:   451	Loss:    22.6983
+Iteration:     1	Loss:     22.245
+Iteration:    51	Loss:    12.0107
julia
sol, us = ude_test(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), trained_ude.ps, trained_ude.st)[1];
+plot_dynamics(sol, us, 0.0:0.01:8.0)

Now that the system is in a better behaved part of parameter space, we return to the original loss function to finish the optimization:

julia
function train_model_2(stateful_ude::StatefulLuxLayer, ts_)
+    ts = collect(ts_)
+
+    function loss_adjoint(θ)
+        x, us = stateful_ude(([-4.0, 0.0], ts, Val(false)), θ)
+        return mean(abs2, 4 .- x[1, :]) .+ 2 * mean(abs2, x[2, :]) .+ mean(abs2, us)
+    end
+
+    callback = function (state, l)
+        state.iter % 10 == 1 && @printf "Iteration: %5d\tLoss: %10g\n" state.iter l
+        return false
+    end
+
+    optf = OptimizationFunction((x, p) -> loss_adjoint(x), AutoZygote())
+    optprob = OptimizationProblem(optf, stateful_ude.ps)
+    res2 = solve(optprob, LBFGS(); callback, maxiters=100)
+
+    return StatefulLuxLayer{true}(stateful_ude.model, res2.u, stateful_ude.st)
+end
+
+trained_ude = train_model_2(trained_ude, 0.0:0.01:8.0)
┌ Warning: Lux.apply(m::AbstractLuxLayer, x::AbstractArray{<:ReverseDiff.TrackedReal}, ps, st) input was corrected to Lux.apply(m::AbstractLuxLayer, x::ReverseDiff.TrackedArray}, ps, st).
+│ 
+│ 1. If this was not the desired behavior overload the dispatch on `m`.
+│ 
+│ 2. This might have performance implications. Check which layer was causing this problem using `Lux.Experimental.@debug_mode`.
+└ @ LuxCoreArrayInterfaceReverseDiffExt ~/.julia/packages/LuxCore/8mVob/ext/LuxCoreArrayInterfaceReverseDiffExt.jl:10
+Iteration:     1	Loss:    12.5986
+Iteration:    11	Loss:     12.559
+Iteration:    21	Loss:     12.471
+Iteration:    31	Loss:    12.4164
+Iteration:    41	Loss:     12.401
+Iteration:    51	Loss:    12.3783
+Iteration:    61	Loss:    12.3635
+Iteration:    71	Loss:    12.3562
+Iteration:    81	Loss:    12.3519
+Iteration:    91	Loss:    12.3471
julia
sol, us = ude_test(([-4.0, 0.0], 0.0:0.01:8.0, Val(true)), trained_ude.ps, trained_ude.st)[1];
+plot_dynamics(sol, us, 0.0:0.01:8.0)

Symbolic Regression ​

Ok so now we have a trained neural network that solves the optimal control problem. But can we replace Dense(4 => 4, gelu) with a symbolic expression? Let's try!

Data Generation for Symbolic Regression ​

First, we need to generate data for the symbolic regression.

julia
ts = reshape(collect(0.0:0.1:8.0), 1, :)
+
+X_train = mlp[1](ts, trained_ude.ps.mlp.layer_1, trained_ude.st.mlp.layer_1)[1]
4×81 Matrix{Float64}:
+ -0.0397511  -0.0317854  -0.0250462  -0.0194458  -0.014873   -0.0112035   -0.00830924   -0.00606573   -0.00435673  -0.00307771   -0.00213748   -0.00145881  -0.000977957  -0.00064367  -0.00041574  -0.000263381  -0.000163581  -9.95524e-5  -5.93359e-5   -3.46184e-5   -1.97604e-5   -1.10295e-5   -6.01676e-6   -3.20614e-6   -1.66797e-6   -8.46734e-7   -4.19207e-7   -2.02301e-7   -9.51099e-8   -4.35387e-8   -1.9396e-8    -8.40435e-9   -3.54008e-9   -1.44879e-9   -5.75761e-10  -2.2207e-10   -8.30827e-11  -3.01347e-11  -1.05906e-11  -3.60445e-12  -1.18735e-12  -3.78357e-13  -1.16566e-13  -3.47018e-14  -9.977e-15  -2.76872e-15  -7.41222e-16  -1.91325e-16  -4.75889e-17  -1.14002e-17  -2.62879e-18  -5.83165e-19  -1.24389e-19  -2.54967e-20  -5.01952e-21  -9.48579e-22  -1.7198e-22   -2.98976e-23  -4.98088e-24  -7.94783e-25  -1.21401e-25  -1.77414e-26  -2.47914e-27  -3.31074e-28  -4.22297e-29  -5.14207e-30  -5.97371e-31  -6.61753e-32  -6.98637e-33  -7.0254e-34   -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0
+  0.111436    0.0419072  -0.017267   -0.0660352  -0.104617   -0.13349     -0.153362     -0.165131     -0.169837    -0.168603     -0.162587     -0.152923    -0.140682     -0.126833    -0.112217    -0.0975332    -0.0833352    -0.0700333   -0.057906     -0.0471162    -0.0377292    -0.0297327    -0.0230565    -0.0175905    -0.0132002    -0.00974017   -0.00706458   -0.00503462   -0.00352389   -0.00242131   -0.00163244   -0.00107935   -0.000699517  -0.000444123  -0.000276081  -0.000167939  -9.99068e-5   -5.8092e-5    -3.29957e-5   -1.82961e-5   -9.89833e-6   -5.2216e-6    -2.68423e-6   -1.34384e-6   -6.5482e-7  -3.10366e-7   -1.43001e-7   -6.40098e-8   -2.78182e-8   -1.17305e-8   -4.7967e-9    -1.90078e-9   -7.29484e-10  -2.70971e-10  -9.73601e-11  -3.38157e-11  -1.13465e-11  -3.67571e-12  -1.14889e-12  -3.46263e-13  -1.00565e-13  -2.81272e-14  -7.57134e-15  -1.96024e-15  -4.87825e-16  -1.16617e-16  -2.67624e-17  -5.89225e-18  -1.24381e-18  -2.51577e-19  -4.87251e-20  -9.03078e-21  -1.60071e-21  -2.71169e-22  -4.38764e-23  -6.77655e-24  -9.98384e-25  -1.40224e-25  -1.87631e-26  -2.39039e-27  -2.89761e-28
+ -0.110793   -0.0822512  -0.057209   -0.0373554  -0.0229031  -0.0131703   -0.00708815   -0.00355998   -0.00166277  -0.000719452  -0.000287171  -0.00010528  -3.52903e-5   -1.07663e-5  -2.97538e-6  -7.4137e-7    -1.65756e-7   -3.30953e-8  -5.87257e-9   -9.21623e-10  -1.273e-10    -1.54007e-11  -1.62392e-12  -1.48515e-13  -1.17228e-14  -7.9471e-16   -4.60432e-17  -2.26861e-18  -9.45911e-20  -3.32116e-21  -9.77079e-23  -2.39675e-24  -4.87773e-26  -8.19525e-28  -1.13111e-29  -1.2761e-31   -1.171e-33    -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0        -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0
+ -0.076181   -0.0468178  -0.026027   -0.0130715  -0.0059069  -0.00238742  -0.000856747  -0.000270747  -7.46883e-5  -1.78221e-5   -3.64438e-6   -6.32607e-7  -9.23267e-8   -1.12204e-8  -1.12451e-9  -9.20332e-11  -6.09117e-12  -3.2282e-13  -1.35657e-14  -4.47562e-16  -1.14788e-17  -2.26603e-19  -3.40918e-21  -3.87018e-23  -3.28239e-25  -2.05921e-27  -9.46098e-30  -3.15184e-32  -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0        -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0          -0.0

This is the training input data. Now we generate the targets

julia
Y_train = mlp[2](X_train, trained_ude.ps.mlp.layer_2, trained_ude.st.mlp.layer_2)[1]
4×81 Matrix{Float64}:
+  0.0516886   0.045542   0.0400219   0.0359023   0.0332579   0.0317798   0.0310197   0.0305495   0.0300455   0.0293137   0.0282779   0.0269476   0.0253838   0.0236698   0.0218909   0.0201231   0.0184276   0.0168498   0.0154194   0.0141528   0.0130554   0.0121237   0.011348    0.0107144   0.0102065   0.00980671   0.00949794   0.00926388   0.00908979   0.0089628   0.00887198   0.00880831   0.0087646   0.00873521   0.00871587   0.00870343   0.0086956   0.00869079   0.0086879   0.00868621   0.00868524   0.00868471   0.00868441   0.00868426   0.00868418   0.00868414   0.00868412   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411   0.00868411
+ -0.166848   -0.15062   -0.130688   -0.112766   -0.0986722  -0.0884998  -0.081764   -0.0778875  -0.0763561  -0.0767414  -0.0786804  -0.0818489  -0.0859417  -0.0906647  -0.0957383  -0.100907   -0.105953   -0.110702   -0.115033   -0.118875   -0.122199   -0.125009   -0.127337   -0.129228   -0.130735   -0.131915    -0.132823    -0.133509    -0.134017    -0.134388   -0.134652    -0.134837    -0.134964   -0.135049    -0.135105    -0.135142    -0.135164   -0.135178    -0.135187   -0.135191    -0.135194    -0.135196    -0.135197    -0.135197    -0.135197    -0.135197    -0.135197    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198    -0.135198
+  1.15297     1.25565    1.33772     1.40111     1.44858     1.4827      1.50547     1.51843     1.52286     1.51998     1.5111      1.49754     1.48062     1.4616      1.44157     1.42147     1.40205     1.38386     1.36727     1.35251     1.33967     1.32874     1.31961     1.31213     1.30613     1.3014       1.29774      1.29496      1.2929       1.29139     1.29031      1.28956      1.28904     1.28869      1.28846      1.28831      1.28822     1.28816      1.28813     1.28811      1.28809      1.28809      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808      1.28808
+  0.884722    0.264828  -0.0595685  -0.162055   -0.165213   -0.142734   -0.121854   -0.10885    -0.103638   -0.10493    -0.111445   -0.121894   -0.134731   -0.148027   -0.159607   -0.167429   -0.170034   -0.166869   -0.158333   -0.145568   -0.130106   -0.113513   -0.0971338  -0.0819568  -0.0685907  -0.0573086   -0.0481281   -0.040896    -0.0353637   -0.0312451  -0.0282564   -0.02614     -0.0246766  -0.0236881   -0.0230357   -0.0226149   -0.0223499  -0.0221869   -0.022089   -0.0220317   -0.0219989   -0.0219807   -0.0219708   -0.0219655   -0.0219629   -0.0219615   -0.0219609   -0.0219605   -0.0219604   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603   -0.0219603

Fitting the Symbolic Expression ​

We will follow the example from SymbolicRegression.jl docs to fit the symbolic expression.

julia
srmodel = MultitargetSRRegressor(;
+    binary_operators=[+, -, *, /], niterations=100, save_to_file=false);

One important note here is to transpose the data because that is how MLJ expects the data to be structured (this is in contrast to how Lux or SymbolicRegression expects the data)

julia
mach = machine(srmodel, X_train', Y_train')
+fit!(mach; verbosity=0)
+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),
+    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_2 = WrappedFunction(stack1),
+        ),
+    ),
+    layer_3 = Dense(4 => 1),            # 5 parameters
+)         # Total: 34 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,
+        layer_2=Lux.initialparameters(rng, hybrid_mlp[2]),
+        layer_3=trained_ude.ps.mlp.layer_3))
+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
+InteractiveUtils.versioninfo()
+
+if @isdefined(MLDataDevices)
+    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
+        println()
+        CUDA.versioninfo()
+    end
+
+    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
+        println()
+        AMDGPU.versioninfo()
+    end
+end
Julia Version 1.11.3
+Commit d63adeda50d (2025-01-21 19:42 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  JULIA_NUM_THREADS = 1
+  JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
+  JULIA_PKG_PRECOMPILE_AUTO = 0
+  JULIA_DEBUG = Literate

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