Deep Learning Models for Global Coordinate Transformations that Linearize PDEs

Deep Learning Models for Global Coordinate Transformations that Linearize PDEs Craig Gin 1, Bethany Lusch 2, Steven L. Brunton 1,3, and J. Nathan Kutz 1 1 Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, USA 2 Argonne Leadership Computing Facility, Argonne National Laboratory, Lemont, IL, USA 3 Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA (Received 11 November 2019) We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE. Our architecture is motivated by the linearizing 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 interpretable, linearizing transforms for nonlinear PDEs. Key Words: Koopman theory, deep neural nets, residual networks, linearizing transforms, Cole-Hopf transform 2010 Mathematics Subject Classification: 35A22, 35A35, 37M99, 65P99, 68T99 1 Introduction Partial differential equations (PDEs) provide a theoretical framework for modeling spatiotemporal systems across the biological, physical and engineering sciences. Analytic solution techniques are readily available for PDEs that are linear and have constant coefficients [12]. These PDEs include canonical models such as the heat equation, wave equation and Laplace's equation which are amenable to standard separation of variable techniques and linear superposition. In contrast, there is no general mathematical architecture for solving nonlinear PDEs as methods like separation of variables fail to hold, thus recourse to computational solutions is necessary. There are a few, but notable, exceptions: (i) the Cole-Hopf transformation [14, 6] for solving diffusively regularized Burgers equation, and (ii) the Inverse Scattering Transform (IST) [1] for solving a class of completely integrable PDEs such as Korteweg deVries (KdV), nonlinear Schr odinger arXiv:1911.02710v1 A deep autoencoder is used to find coordinate transformations to linearize PDEs. The encoder finds a set of intrinsic coordinates for which the dynamics are linear.