Deep learning models for global coordinate transformations that linearise PDEs

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.