Journal
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
Volume 32, Issue 3, Pages 515-539Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0956792520000327
Keywords
Koopman theory; deep neural networks; residual networks; linearising transforms; Cole– Hopf transform
Categories
Funding
- Defense Advanced Research Projects Agency [DARPA PA-18-01-FP-125]
- Army Research Office [ARO W911NF-17-1-0306]
- Air Force Office of Scientific Research [FA9550-17-1-0329]
- STF at the University of Washington
- Argonne Leadership Computing Facility
- DOE Office of Science User Facility [DE-AC02-06CH11357]
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The study developed a deep autoencoder architecture that transforms non-linear PDEs into linear PDEs by utilizing a residual network architecture and intrinsic coordinates encoding. The resulting dynamics are given by a Koopman operator matrix, allowing for multiple time step prediction. Demonstrations on various examples show the robustness of the method in discovering linearizing transforms for non-linear PDEs.
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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