A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks

Abstract

Physics-informed neural networks (PINNs) often struggle to resolve shocks because smooth neural representations and globally averaged residual losses tend to underemphasize localized high-gradient regions. We introduce a residual-tracked Gaussian weighting strategy that represents the spatial distribution of the PDE-residual loss through a small number of learnable parameters, without resampling the collocation points or prescribing the shock location. The Gaussian center and width are updated from the evolving PDE residual, allowing the weighting field to follow stationary and moving shocks during training without prior knowledge of their positions or trajectories. The method is assessed using one-dimensional low-viscosity Burgers problems and a two-dimensional Euler Riemann problem with four interacting shocks. For $\nu =5\times 10^{-4}$, the relative $L_2$ error decreases from approximately $45\%$ to $13\%$ for the stationary shock and from $33\%$ to $14\%$ for the moving shock, compared with a standard PINN. In the two-dimensional problem, the maximum absolute density error decreases from $1.08$ to $0.62$. These results show that a compact, dynamically localized residual weighting can substantially improve PINN resolution of shock-dominated solutions while retaining the original collocation set and avoiding prescribed shock trajectories.