{"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/numerical-gaussian-processes-for-time","title":"Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations","arxiv_id":"1703.10230","date":"2017-03-29","proceeding":null,"authors":["Maziar Raissi","Paris Perdikaris","George Em. Karniadakis"],"abstract":"We introduce the concept of numerical Gaussian processes, which we define as\nGaussian processes with covariance functions resulting from temporal\ndiscretization of time-dependent partial differential equations. Numerical\nGaussian processes, by construction, are designed to deal with cases where: (1)\nall we observe are noisy data on black-box initial conditions, and (2) we are\ninterested in quantifying the uncertainty associated with such noisy data in\nour solutions to time-dependent partial differential equations. Our method\ncircumvents the need for spatial discretization of the differential operators\nby proper placement of Gaussian process priors. This is an attempt to construct\nstructured and data-efficient learning machines, which are explicitly informed\nby the underlying physics that possibly generated the observed data. The\neffectiveness of the proposed approach is demonstrated through several\nbenchmark problems involving linear and nonlinear time-dependent operators. In\nall examples, we are able to recover accurate approximations of the latent\nsolutions, and consistently propagate uncertainty, even in cases involving very\nlong time integration.","url_abs":"http://arxiv.org/abs/1703.10230v1","url_pdf":"http://arxiv.org/pdf/1703.10230v1.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":"numerical-gaussian-processes-for-time","repo_url":"https://github.com/maziarraissi/NumericalGP","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"gaussian-processes","task_name":"Gaussian Processes"}],"methods":[{"method_slug":"gaussian-process","method_name":"Gaussian Process"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1703.10230","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}