{"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/hidden-physics-models-machine-learning-of","title":"Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations","arxiv_id":"1708.00588","date":"2017-08-02","proceeding":null,"authors":["Maziar Raissi","George Em. Karniadakis"],"abstract":"While there is currently a lot of enthusiasm about \"big data\", useful data is\nusually \"small\" and expensive to acquire. In this paper, we present a new\nparadigm of learning partial differential equations from {\\em small} data. In\nparticular, we introduce \\emph{hidden physics models}, which are essentially\ndata-efficient learning machines capable of leveraging the underlying laws of\nphysics, expressed by time dependent and nonlinear partial differential\nequations, to extract patterns from high-dimensional data generated from\nexperiments. The proposed methodology may be applied to the problem of\nlearning, system identification, or data-driven discovery of partial\ndifferential equations. Our framework relies on Gaussian processes, a powerful\ntool for probabilistic inference over functions, that enables us to strike a\nbalance between model complexity and data fitting. The effectiveness of the\nproposed approach is demonstrated through a variety of canonical problems,\nspanning a number of scientific domains, including the Navier-Stokes,\nSchr\\\"odinger, Kuramoto-Sivashinsky, and time dependent linear fractional\nequations. The methodology provides a promising new direction for harnessing\nthe long-standing developments of classical methods in applied mathematics and\nmathematical physics to design learning machines with the ability to operate in\ncomplex domains without requiring large quantities of data.","url_abs":"http://arxiv.org/abs/1708.00588v2","url_pdf":"http://arxiv.org/pdf/1708.00588v2.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":"hidden-physics-models-machine-learning-of","repo_url":"https://github.com/maziarraissi/HPM","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"machine-learning","task_name":"BIG-bench Machine Learning"},{"task_slug":"gaussian-processes","task_name":"Gaussian Processes"},{"task_slug":"small-data","task_name":"Small Data Image Classification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1708.00588","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}