In this talk, we'll discuss two frameworks for learning PDEs from data. The first, Modal Operator Regression for Physics (MOR-Physics), fits spatial operators by parameterizing them as compositions of point-wise applied neural-networks and pseudo-differential operators with neural network symbols. This parameterization allows for the introduction of physically motivated inductive bias such as conservation. We verify that the method can recover the spatial operators from solutions to PDEs and that inductive bias improves generalization error. Additionally, we demonstrate that the method can extract a continuum scale model from molecular dynamics simulations of colloidal Poiseuille flow.
The second framework, Control Volume Physics-Informed Neural Networks (CV-PINNs) is a variant of PINNs designed for PDEs with less regular solutions, particularly nonlinear hyperbolic PDEs. As a forward solver, CV-PINNs is able to solve Riemann problems for various hyperbolic PDEs without the heavy regularization needed for standard PINNs. For inverse problems with the Euler equations, CV-PINNs is able to extract equations of state (EOS) from DSMC simulations of Sod shocks. With additional regularization, CV-PINNs maintains the thermodynamic consistency of the EOS and therefore the hyperbolicity of the Euler equations.