2024/03/14 by Tim De Ryck, Siddhartha Mishra, Roberto Molinaro
Physics and Astronomy · Engineering · Decision Sciences · #Model Reduction and Neural Networks #Fluid Dynamics and Turbulent Flows #Probabilistic and Robust Engineering Design
paper · doi:10.1137/22m1522504
.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.KeywordsPINNshyperbolic conservation lawsdeep learningMSC codes65M99