Variational Formulations for Solving PDEs with Non-Smooth Solutions using Non-Linear Surrogates

This talk intends to address the challenge of solving Partial Differential Equations (PDEs) with smooth
or non-smooth solutions by formulating variational PDE formulations resulting in a soft-constrained
optimization problem. The flexibility of the variational formulation enables us to use hybrid non-linear
surrogates to approximate discontinuous shocks while solving forward or inverse PDE problems. We
first explore general concepts and tools necessary for solving PDEs under a variational formulation with
general non-linear surrogates and boundary conditions. We then compare the numerical performance of
Physics Informed Neural Networks (PINNs) as surrogates against Polynomial Surrogate Models (PSMs).
Our goal is to open up the discussion regarding the class of problems that genuinely require the use of
Neural Networks. Our findings indicate that PSMs outperform PINNs by several orders of magnitude in
both accuracy and runtime. Furthermore, we introduce a new method for approximating discontinuous
functions using modified global spectral methods. We extend this method to solve PDEs with nonsmooth
solutions, providing an innovative solution to a highly challenging problem.