Deep Learning Models for Global Coordinate Transformations that Linearize PDEs

Abstract

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 $\mathbf{K}$. 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 $\mathbf{K}$ 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.

arXiv comment23 pages, 18 figures

References in corpus (12)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Deep learning for universal linear embeddings of nonlinear dynamics
• Data-driven discovery of coordinates and governing equations
• Understanding Deep Convolutional Networks
• Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
• VAMPnets: Deep learning of molecular kinetics
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
• Data-driven model reduction and transfer operator approximation
• Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability
• Geometry of the ergodic quotient reveals coherent structures in flows
• Koopman analysis of Burgers equation

Cited by in corpus (8)

• Artificial Neural Networks for Photonic Applications: From Algorithms to Implementation
• An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes
• Manifold embedding data-driven mechanics
• Neural network representability of fully ionized plasma fluid model closures
• Fast Dynamic 1D Simulation of Divertor Plasmas with Neural PDE Surrogates
• Identification of high order closure terms from fully kinetic simulations using machine learning
• MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations
• Transformer-based Koopman Autoencoder for Linearizing Fisher's Equation