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

[2606.26013] A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks Skip to main content Search Submit Donate Log in Physics > Fluid Dynamics arXiv:2606.26013 (physics) [Submitted on 24 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks Authors: Kheir-eddine Otmani , Abdelhalim Azzouz , Nourelhouda Groun , Esteban Ferrer View PDF HTML (experimental) 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 $\nu=5\times10^{-4}$, 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. Subjects: Fluid Dynamics (physics.flu-dyn) Cite as: arXiv:2606.26013 [physics.flu-dyn] (or arXiv:2606.26013v2 [physics.flu-dyn] for this version) https://doi.org/10.48550/arXiv.2606.26013 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Kheir-Eddine Otmani [ view email ] [v1] Wed, 24 Jun 2026 16:32:51 UTC (3,952 KB) [v2] Thu, 27 Aug 2026 21:30:16 UTC (4,539 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: physics.flu-dyn < prev | next > new | recent | 2026-06 Change to browse by: physics References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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