Characteristic Based Physics Informed Neural Networks for Advection Equations with a Focus on Discontinuous Solutions

Abstract

This paper presents a characteristic-based loss function for physics-informed neural networks (PINNs) that circumvents automatic differentiation and significantly reduces training cost for advection equations. However, standard PINNs often fail to accurately resolve discontinuous solutions, as their global smoothness prior and spectral bias lead to smeared shocks or spurious oscillations near sharp gradients. To accurately capture these discontinuities, the density of sampling points in their vicinity must be sufficiently high, and an appropriate sampling strategy must be employed. For problems involving discontinuous initial and boundary conditions, several complementary techniques are introduced: an adaptive sampling strategy that concentrates collocation points along characteristic paths, a Fourier feature mapping with two-stage training and adaptive weighting to mitigate spectral bias, and an adaptive median filter applied to the spatial data at each time instant, which suppresses spurious oscillations while preserving sharp features such as corners and peaks. The effectiveness of the proposed framework is demonstrated through numerical experiments, showing improved accuracy and reduced computational effort in approximating discontinuous solutions.