{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/solving-partial-differential-equations-with-4","title":"Solving partial differential equations with sampled neural networks","arxiv_id":"2405.20836","date":"2024-05-31","proceeding":null,"authors":["Chinmay Datar","Taniya Kapoor","Abhishek Chandra","Qing Sun","Iryna Burak","Erik Lien Bolager","Anna Veselovska","Massimo Fornasier","Felix Dietrich"],"abstract":"Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. Using neural networks as an ansatz for the solution has proven a challenge in terms of training time and approximation accuracy. In this contribution, we discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges. In most examples, the random sampling schemes outperform iterative, gradient-based optimization of physics-informed neural networks regarding training time and accuracy by several orders of magnitude. For time-dependent PDE, we construct neural basis functions only in the spatial domain and then solve the associated ordinary differential equation with classical methods from scientific computing over a long time horizon. This alleviates one of the greatest challenges for neural PDE solvers because it does not require us to parameterize the solution in time. For second-order elliptic PDE in Barron spaces, we prove the existence of sampled networks with $L^2$ convergence to the solution. We demonstrate our approach on several time-dependent and static PDEs. We also illustrate how sampled networks can effectively solve inverse problems in this setting. Benefits compared to common numerical schemes include spectral convergence and mesh-free construction of basis functions.","url_abs":"https://arxiv.org/abs/2405.20836v1","url_pdf":"https://arxiv.org/pdf/2405.20836v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"solving-partial-differential-equations-with-4","repo_url":"https://gitlab.com/felix.dietrich/swimpde","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"solving-partial-differential-equations-with-4","repo_url":"https://gitlab.com/felix.dietrich/swimpde-paper","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2405.20836","atlas_url":"https://app.syntology.ai/?focus=2405.20836","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}