Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations

TL;DR

This paper introduces Lagrangian Gaussian Processes (LGPs) that learn dynamical systems from position data while preserving geometric structure, ensuring physically consistent, stable long-term predictions with uncertainty quantification.

Contribution

The paper presents a novel LGP framework that learns from position data using discrete Euler-Lagrange equations, maintaining physical structure and enabling stable long-term predictions.

Findings

LGPs effectively learn physically consistent dynamics from sparse position data.

The method demonstrates superior stability and generalization in synthetic and real-world case studies.

LGPs provide uncertainty quantification without requiring velocity or momentum data.

Abstract

In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert principle, which governs the motion of dynamical systems, is preserved by construction in the absence of external forces. This allows learning physically consistent models that overcome erroneous drift in the system's energy, thereby providing stable long-term predictions. At the core of our approach lie linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations and variational discretization schemes. Thereby and unlike prior work, the method enables learning dynamics from discrete position snapshots, i.e., without access to a system's velocities or momenta. This is particularly relevant for a large class of practical scenarios where only position measurements are available, for instance, in motion capture or visual servoing applications. We demonstrate the data-efficiency and generalization capabilities of the LGPs in various synthetic and real-world case studies, including a real-world soft robot with hysteresis. The experimental results underscore that the LGPs learn physically consistent dynamics with uncertainty quantification solely from sparse positional data and enable stable long-term predictions.