Learning Dynamical Systems with the Spectral Exterior Calculus

TL;DR

This paper introduces a spectral exterior calculus framework for learning dynamical systems on Riemannian manifolds, using eigenfunctions of the Laplacian to accurately reconstruct vector fields from data.

Contribution

It develops a novel data-driven method that leverages spectral exterior calculus to approximate dynamical systems on manifolds, with proven convergence and numerical validation.

Findings

Accurate reconstruction of vector fields on manifolds.

Convergence of the method as data size increases.

Numerical examples on the circle and torus show good agreement with true dynamics.

Abstract

We present a data-driven framework for learning dynamical systems on compact Riemannian manifolds based on the spectral exterior calculus (SEC). This approach represents vector fields as linear combinations of frame elements constructed using the eigenfunctions of the Laplacian on smooth functions, along with their gradients. Such reconstructed vector fields generate dynamical flows that consistently approximate the true system, while being compatible with the nonlinear geometry of the manifold. The data-driven implementation of this framework utilizes embedded data points and tangent vectors as training data, along with a graph-theoretic approximation of the Laplacian. In this paper, we prove the convergence of the SEC-based reconstruction in the limit of large data. Moreover, we illustrate the approach numerically with applications to dynamical systems on the unit circle and the 2-torus. In these examples, the reconstructed vector fields compare well with the true vector fields, in terms of both pointwise estimates and generation of orbits.