Learning dynamics on invariant measures using PDE-constrained optimization

TL;DR

This paper introduces a PDE-constrained optimization framework to learn continuous-time dynamical systems from invariant measures, enabling stable predictions and uncertainty quantification, demonstrated on classical and real-world systems.

Contribution

It reformulates the inverse problem of learning ODEs/SDEs from data as a PDE-constrained optimization, improving stability and quantification capabilities.

Findings

Effective on Van der Pol oscillator and Lorenz-63 system

Successful application to Hall-effect thruster dynamics

Enhanced stability over direct trajectory simulation

Abstract

We extend the methodology in [Yang et al., 2023] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.