Parameter Estimation formulation in PINNs

[Vaibhavdixit02]

I have been confused about the parameter estimation of differential equation parameters here for a while now.

I think the current formulation of the problem is suboptimal. The loss function only takes into account the error of the neural network surrogate and data and doesn't utilize a loss function to capture the norm of the physics equation solution at the current parameters which is the typical formulation of this without a NN solver. I think this will improve the training of the surrogate as well since now both the data and physics equation would be converging.

There are two parts to this, which I think can be done independently:

1. if we only consider initial value problems and ignore the boundary conditions data and losses it could be added as a collocation loss and solved as an OptimizationProblem.
   It could be a nested solve to a loose tolerance or the losses added up together.

2. The case with boundary conditions could be formulated as a boundary value problem. Though that makes it a harder problem we have pretty fast solvers now.

[Vaibhavdixit02]

This translates over to the bayesian PINNs, where neither https://arxiv.org/pdf/2205.08304.pdf or https://arxiv.org/pdf/2003.06097.pdf mention the formulation of the likelihood for the parameter estimation case clearly.

[AstitvaAggarwal]

How would the derivatives be calculated in cases where an equation contains multiple different differential operator terms? if im understanding it correctly you forward solve at each updated parameter p value? or is it a loss from collocation of data and the updated parameter values which is added with total loss?

[Vaibhavdixit02]

I think the collocation loss would be more efficient and generalize to PDEs

[AstitvaAggarwal]

Okay so you would take derivatives of the interpolations in that case?

[Vaibhavdixit02]

I guess. But based on the above papers it looks like it is assumed that the data for the derivatives is sometimes directly available so I would use that if it's available