Structure-preserving Gaussian Process Dynamics

TL;DR

This paper introduces a novel method combining Gaussian processes with structure-preserving numerical integrators, enabling learning of dynamical systems that respect physical invariants.

Contribution

It develops an implicit layer for GP regression integrated with variational inference to incorporate structure-preserving integrators in model learning.

Findings

Successfully preserves invariants like energy and volume in learned models

Enables accurate long-term predictions of physical systems

Bridges Gaussian process modeling with classical numerical methods

Abstract

Most physical processes posses structural properties such as constant energies, volumes, and other invariants over time. When learning models of such dynamical systems, it is critical to respect these invariants to ensure accurate predictions and physically meaningful behavior. Strikingly, state-of-the-art methods in Gaussian process (GP) dynamics model learning are not addressing this issue. On the other hand, classical numerical integrators are specifically designed to preserve these crucial properties through time. We propose to combine the advantages of GPs as function approximators with structure preserving numerical integrators for dynamical systems, such as Runge-Kutta methods. These integrators assume access to the ground truth dynamics and require evaluations of intermediate and future time steps that are unknown in a learning-based scenario. This makes direct inference of the GP dynamics, with embedded numerical scheme, intractable. Our key technical contribution is the evaluation of the implicitly defined Runge-Kutta transition probability. In a nutshell, we introduce an implicit layer for GP regression, which is embedded into a variational inference-based model learning scheme.