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Deep learning models for global coordinate transformations that linearise PDEs

Published online by Cambridge University Press:  24 September 2020

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
Corresponding

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.

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Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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