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Solving and learning nonlinear PDEs with Gaussian processes

Chen, Yifan and Hosseini, Bamdad and Owhadi, Houman and Stuart, Andrew M. (2021) Solving and learning nonlinear PDEs with Gaussian processes. Journal of Computational Physics, 447 . Art. No. 110668. ISSN 0021-9991. doi:10.1016/j.jcp.2021.110668. https://resolver.caltech.edu/CaltechAUTHORS:20210719-210146136

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Abstract

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss–Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jcp.2021.110668DOIArticle
https://arxiv.org/abs/2103.12959arXivDiscussion Paper
ORCID:
AuthorORCID
Owhadi, Houman0000-0002-5677-1600
Stuart, Andrew M.0000-0001-9091-7266
Additional Information:© 2021 Elsevier Inc. Available online 31 August 2021. The authors gratefully acknowledge support by the Air Force Office of Scientific Research under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). HO also acknowledges support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning). YC is also grateful for support from the Caltech Kortchak Scholar Program. CRediT authorship contribution statement: Yifan Chen: Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Bamdad Hosseini: Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Houman Owhadi: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Writing – original draft, Writing – review & editing. Andrew M. Stuart: Formal analysis, Methodology, Writing – review & editing. Declaration of Competing Interest: George Karniadakis, Houman Owhadi and Andrew M Stuart are CoPIs on a MURI grant (number 19) awarded by AFOSR. We do not think that this constitutes a conflict of interest, but we are disclosing this for the sake of full transparency and to prevent ambiguity.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-20-1-0358
Air Force Office of Scientific Research (AFOSR)FA9550-18-1-0271
Kortschak Scholars ProgramUNSPECIFIED
Subject Keywords:Kernel methods; Gaussian processes; Nonlinear partial differential equations; Inverse problems; Optimal recovery
Classification Code:AMS subject classifications. 60G15, 65M75, 65N75, 65N35, 47B34, 41A15, 35R30, 34B15.
DOI:10.1016/j.jcp.2021.110668
Record Number:CaltechAUTHORS:20210719-210146136
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210719-210146136
Official Citation:Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart, Solving and learning nonlinear PDEs with Gaussian processes, Journal of Computational Physics, Volume 447, 2021, 110668, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2021.110668.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109921
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:19 Jul 2021 22:26
Last Modified:22 Sep 2021 19:39

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