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Deep learning models for global coordinate transformations that linearise PDEs

Part of: Numerical problems in dynamical systems Partial differential equations Approximation methods and numerical treatment of dynamical systems Artificial intelligence (68Txx)

Published online by Cambridge University Press:  24 September 2020

CRAIG GIN Open the ORCID record for CRAIG GIN [Opens in a new window] ,
BETHANY LUSCH Open the ORCID record for BETHANY LUSCH [Opens in a new window] ,
STEVEN L. BRUNTON  and
J. NATHAN KUTZ Open the ORCID record for J. NATHAN KUTZ [Opens in a new window]
CRAIG GIN
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA98195, USA emails: crgin@ncsu.edu; kutz@uw.edu
BETHANY LUSCH
Affiliation:
Argonne Leadership Computing Facility, Argonne National Laboratory, Lemont, IL60439, USA email: blusch@anl.gov
STEVEN L. BRUNTON
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA98195, USA emails: crgin@ncsu.edu; kutz@uw.edu Department of Mechanical Engineering, University of Washington, Seattle, WA98195, USA email: sbrunton@uw.edu
J. NATHAN KUTZ
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA98195, USA emails: crgin@ncsu.edu; kutz@uw.edu
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Abstract

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

Keywords

Koopman theorydeep neural networksresidual networkslinearising transformsCole–Hopf transform

MSC classification

Primary: 35A22: Transform methods (e.g. integral transforms)
Secondary: 35A35: Theoretical approximation to solutions 37M99: None of the above, but in this section 65P99: None of the above, but in this section 68T99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 515 - 539
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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