Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations

2026-05-07

AI Summary

This work addresses the challenge of learning physically consistent and long-term stable data-driven dynamical models from sparse positional observations alone. It proposes the Lagrangian Gaussian Process (LGP), which leverages a discrete forced Euler–Lagrange equation combined with a variational discretization scheme. The method exactly preserves the geometric structure of the Lagrange–d’Alembert principle in the absence of external forces, enabling structure-preserving modeling without requiring velocity or momentum measurements. Furthermore, LGP inherently supports uncertainty quantification. As the first approach to learn structure-preserving dynamics exclusively from discrete position data, LGP demonstrates superior data efficiency, generalization capability, and long-term prediction stability across multiple synthetic and real-world systems, including soft robotic platforms exhibiting hysteresis.

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.

Problem

• dynamics learning
• position-only data
• energy conservation
• long-term prediction
• physical consistency

Innovation

• Lagrangian Gaussian Processes
• Structure Preservation
• Discrete Euler-Lagrange Equations
• Data-Efficient Dynamics Learning
• Position-Only Learning