Inverse Problems for Partial Differential Equations with Jump Discontinuities in Coefficients via Two-Stage Physics-Informed Deep Learning and Statistical Mixture Models

[2510.14656] Inverse Problems for Partial Differential Equations with Jump Discontinuities in Coefficients via Two-Stage Physics-Informed Deep Learning and Statistical Mixture Models Skip to main content Search Submit Donate Log in Statistics > Machine Learning arXiv:2510.14656 (stat) [Submitted on 16 Oct 2025 ( v1 ), last revised 18 Aug 2026 (this version, v3)] Title: Inverse Problems for Partial Differential Equations with Jump Discontinuities in Coefficients via Two-Stage Physics-Informed Deep Learning and Statistical Mixture Models Authors: Zhikun Zhang , Guanyu Pan , Xiangjun Wang , Yong Xu , Guangtao Zhang View PDF HTML (experimental) Abstract: This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main network approximates the PDE solution and an auxiliary coefficient sub network provides a relaxed continuous surrogate of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated into the physics residual to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which estimate the number of coefficient regimes and provide heuristic search intervals for coefficient values and candidate transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is represented explicitly as a hard piecewise-constant function over the spatiotemporal domain. Numerical experiments on different PDE types with jump-discontinuous coefficients demonstrate that the proposed framework achieves accurate parameter estimation with acceptable computational costs compared to existing methods. This work provides an effective integrated workflow for inverse problems governed by PDEs with discontinuous parameter structures, particularly in nonstationary and heterogeneous systems. Subjects: Machine Learning (stat.ML) ; Machine Learning (cs.LG) Cite as: arXiv:2510.14656 [stat.ML] (or arXiv:2510.14656v3 [stat.ML] for this version) https://doi.org/10.48550/arXiv.2510.14656 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.jcp.2026.115334 Focus to learn more DOI(s) linking to related resources Submission history From: Zhikun Zhang [ view email ] [v1] Thu, 16 Oct 2025 13:12:26 UTC (25,645 KB) [v2] Sun, 21 Jun 2026 16:11:04 UTC (36,298 KB) [v3] Tue, 18 Aug 2026 01:53:09 UTC (35,624 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: stat.ML < prev | next > new | recent | 2025-10 Change to browse by: cs cs.LG stat 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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