Learning a Reduced Basis of Dynamical Systems using an Autoencoder

TL;DR

This paper introduces an autoencoder with latent space penalization to identify finite-dimensional manifolds underlying PDEs, effectively reducing complex dynamical systems to simpler latent representations.

Contribution

It presents a novel autoencoder-based method with penalization to discover reduced manifolds for PDEs, aligning with known physics and inertial manifold dimensions.

Findings

Latent space for K-S matches inertial manifold dimension

No reduced latent space for KdV, consistent with infinite-dimensional dynamics

Active dimensions decrease with increased damping in damped KdV

Abstract

Machine learning models have emerged as powerful tools in physics and engineering. Although flexible, a fundamental challenge remains on how to connect new machine learning models with known physics. In this work, we present an autoencoder with latent space penalization, which discovers finite dimensional manifolds underlying the partial differential equations of physics. We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations. We show that the resulting optimal latent space of the K-S equation is consistent with the dimension of the inertial manifold. The results for the KdV equation imply that there is no reduced latent space, which is consistent with the truly infinite dimensional dynamics of the KdV equation. In the case of the damped KdV equation, we find that the number of active dimensions decreases with increasing damping coefficient. We then uncover a nonlinear basis representing the manifold of the latent space for the K-S equation.