Training PINN using H^1 Norm

[alanraydan]

Hi all,

I'm using DeepXDE to solve the primitive equations (a collection of 4 PDEs) with two spatial variables and one time variable. Passing the data (PDE residuals and IC/BCs) to dde.data.TimePDE() and training the model with this data is equivalent to minimizing the $L^2$ norm between the PINN and the data; however, I am interested in minimizing the $H^1$ norm as well.

For the PDE residuals this isn't a problem since I can just pass the Jacobian of the PDE as another residual, e.g.

h1_pde_res = dde.grad.jacobian(pde_res, x, i=0, j=0)

The periodic BCs also aren't hard because I can just use the argument derivative_order=1. But for the ICs and Dirichlet BCs there is no option to select higher order derivatives.

Mathematically speaking, given initial data $f(x)$, I would like to train my PINN to minimize the $H^1$ residual $(\partial_x (f(x) - PINN_{\theta}(x, t=0) ))^2$ as well as the $L^2$ residual $(f(x) - PINN_{\theta}(x, t=0) )^2$. Likewise for the Dirichlet BC residuals.

MAIN QUESTION: How can I train a PINN to minimize the gradient of the IC residuals and Dirichlet BC residuals rather than just the residuals themselves?

[forxltk]

Maybe you can use Operator BC instead of IC and Dirichlet BC. Or define your own loss function, and use it for loss in model.compile

[alanraydan]

This worked!

[inodozi]

Maybe you can use Operator BC instead of IC and Dirichlet BC. Or define your own loss function, and use it for loss in model.compile

Hi There
Is it possible to have two different types of metrics for the residuals?
For example, L_2 for the PDE residuals and another metric like Wasserstein distance for the Initial conditions?

[forxltk]

Yes, see https://deepxde.readthedocs.io/en/latest/_modules/deepxde/model.html?highlight=loss%20function

[lululxvi]

You may be also interested in gPINN https://doi.org/10.1016/j.cma.2022.114823