G-PINNs: Gaussian-based spatially weighted formulation for PINNs: 1D low-viscous Burgers

Abstract

We introduce a Gaussian-based spatially weighted loss framework (G-PINNs) for physics-informed neural networks (PINNs) to improve the resolution of sharp discontinuities and shock waves. The proposed method dynamically prioritizes collocation points in high-gradient regions during optimization. Without requiring prior knowledge of the shock location or trajectory, the framework can autonomously detect and track moving discontinuities directly from the PDE residual landscape, making it broadly applicable to problems in which the position of shocks or discontinuities is unknown \textit{a priori}. The approach is validated using one-dimensional quasi-inviscid Burgers' problems exhibiting both stationary and moving shock waves. For the low-viscosity regime $(ν= 0.0005)$, the proposed method achieves $L_2$ relative errors of approximately $13\%$ and $14\%$ for the stationary and moving shock cases, respectively, compared with $45\%$ and $33\%$ obtained when using standard PINNs.