Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics

TL;DR

This paper presents a novel data-driven approach to learn the equations of motion for mechanical systems directly from position data, leveraging discrete Lagrangian mechanics to ensure physical consistency and interpretability.

Contribution

The method introduces a way to learn system dynamics from position-only data using discrete Lagrangian principles and neural networks, without needing velocity measurements.

Findings

Successfully models human motion data.

Effectively separates conservative and non-conservative forces.

Preserves symplectic structure in learned dynamics.

Abstract

We introduce a data-driven method for learning the equations of motion of mechanical systems directly from position measurements, without requiring access to velocity data. This is particularly relevant in system identification tasks where only positional information is available, such as motion capture, pixel data or low-resolution tracking. Our approach takes advantage of the discrete Lagrange-d'Alembert principle and the forced discrete Euler-Lagrange equations to construct a physically grounded model of the system's dynamics. We decompose the dynamics into conservative and non-conservative components, which are learned separately using feed-forward neural networks. In the absence of external forces, our method reduces to a variational discretization of the action principle naturally preserving the symplectic structure of the underlying Hamiltonian system. We validate our approach on a variety of synthetic and real-world datasets, demonstrating its effectiveness compared to baseline methods. In particular, we apply our model to (1) measured human motion data and (2) latent embeddings obtained via an autoencoder trained on image sequences. We demonstrate that we can faithfully reconstruct and separate both the conservative and forced dynamics, yielding interpretable and physically consistent predictions.