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 ν=5×10−4, the relative L2 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.
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BibTeX
@misc{otmani2026gaussianbased,
title = {{A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks}},
author = {Kheir-eddine Otmani and Abdelhalim Azzouz and Nourelhouda Groun and Esteban Ferrer},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.26013},
eprint = {2606.26013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks
AU - Kheir-eddine Otmani
AU - Abdelhalim Azzouz
AU - Nourelhouda Groun
AU - Esteban Ferrer
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2606.26013
UR - https://arxiv.org/abs/2606.26013
AB - 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 ν=5×10^−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.
ER -