{"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/inferring-solutions-of-differential-equations","title":"Inferring solutions of differential equations using noisy multi-fidelity data","arxiv_id":"1607.04805","date":"2016-07-16","proceeding":null,"authors":["Maziar Raissi","Paris Perdikaris","George Em. Karniadakis"],"abstract":"For more than two centuries, solutions of differential equations have been\nobtained either analytically or numerically based on typically well-behaved\nforcing and boundary conditions for well-posed problems. We are changing this\nparadigm in a fundamental way by establishing an interface between\nprobabilistic machine learning and differential equations. We develop\ndata-driven algorithms for general linear equations using Gaussian process\npriors tailored to the corresponding integro-differential operators. The only\nobservables are scarce noisy multi-fidelity data for the forcing and solution\nthat are not required to reside on the domain boundary. The resulting\npredictive posterior distributions quantify uncertainty and naturally lead to\nadaptive solution refinement via active learning. This general framework\ncircumvents the tyranny of numerical discretization as well as the consistency\nand stability issues of time-integration, and is scalable to high-dimensions.","url_abs":"http://arxiv.org/abs/1607.04805v1","url_pdf":"http://arxiv.org/pdf/1607.04805v1.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":"inferring-solutions-of-differential-equations","repo_url":"https://github.com/maziarraissi/TutorialGP","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"active-learning","task_name":"Active Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1607.04805","atlas_url":"https://app.syntology.ai/?focus=1607.04805","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}