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Fourier Neural Operator for Parametric Partial Differential Equations

Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima (2020) Fourier Neural Operator for Parametric Partial Differential Equations. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20201106-120140981

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Abstract

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and the Navier-Stokes equation (including the turbulent regime). Our Fourier neural operator shows state-of-the-art performance compared to existing neural network methodologies and it is up to three orders of magnitude faster compared to traditional PDE solvers.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2010.08895arXivDiscussion Paper
ORCID:
AuthorORCID
Kovachki, Nikola0000-0002-3650-2972
Azizzadenesheli, Kamyar0000-0001-8507-1868
Liu, Burigede0000-0002-6518-3368
Bhattacharya, Kaushik0000-0003-2908-5469
Additional Information:Z. Li gratefully acknowledges the financial support from the Kortschak Scholars Program. A. Anandkumar is supported in part by Bren endowed chair, LwLL grants, Beyond Limits, Raytheon, Microsoft, Google, Adobe faculty fellowships, and DE Logi grant. K. Bhattacharya, N. B. Kovachki, B. Liu and A. M. Stuart gratefully acknowledge the financial support of the Army Research Laboratory through the Cooperative Agreement Number W911NF-12-0022. Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
Funders:
Funding AgencyGrant Number
Kortschak Scholars ProgramUNSPECIFIED
Bren Professor of Computing and Mathematical SciencesUNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Learning with Less Labels (LwLL)UNSPECIFIED
Beyond LimitsUNSPECIFIED
Raytheon CompanyUNSPECIFIED
Microsoft Faculty FellowshipUNSPECIFIED
Google Faculty Research AwardUNSPECIFIED
AdobeUNSPECIFIED
Caltech De Logi FundUNSPECIFIED
Army Research LaboratoryW911NF-12-0022
Record Number:CaltechAUTHORS:20201106-120140981
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201106-120140981
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106480
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:06 Nov 2020 20:31
Last Modified:06 Nov 2020 20:31

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