{"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/a-discussion-on-solving-partial-differential","title":"A Discussion on Solving Partial Differential Equations using Neural Networks","arxiv_id":"1904.07200","date":"2019-04-15","proceeding":null,"authors":["Tim Dockhorn"],"abstract":"Can neural networks learn to solve partial differential equations (PDEs)? We\ninvestigate this question for two (systems of) PDEs, namely, the Poisson\nequation and the steady Navier--Stokes equations. The contributions of this\npaper are five-fold. (1) Numerical experiments show that small neural networks\n(< 500 learnable parameters) are able to accurately learn complex solutions for\nsystems of partial differential equations. (2) It investigates the influence of\nrandom weight initialization on the quality of the neural network approximate\nsolution and demonstrates how one can take advantage of this non-determinism\nusing ensemble learning. (3) It investigates the suitability of the loss\nfunction used in this work. (4) It studies the benefits and drawbacks of\nsolving (systems of) PDEs with neural networks compared to classical numerical\nmethods. (5) It proposes an exhaustive list of possible directions of future\nwork.","url_abs":"http://arxiv.org/abs/1904.07200v1","url_pdf":"http://arxiv.org/pdf/1904.07200v1.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":"a-discussion-on-solving-partial-differential","repo_url":"https://github.com/timudk/SPDENN","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[{"task_slug":"ensemble-learning","task_name":"Ensemble Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}