wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws

TL;DR

wPINNs introduce a novel neural network approach that effectively approximates entropy solutions of hyperbolic conservation laws, overcoming regularity limitations of traditional PINNs by leveraging a min-max optimization framework based on Kruzkhov entropies.

Contribution

The paper proposes wPINNs, a new variant of PINNs that accurately approximates entropy solutions of hyperbolic PDEs using a min-max optimization approach with theoretical error bounds.

Findings

wPINNs accurately approximate entropy solutions in numerical experiments.

wPINNs outperform traditional PINNs on discontinuous solutions.

Theoretical error bounds validate the method's effectiveness.

Abstract

Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.