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

TL;DR

The paper introduces a data-driven, structure-preserving approach to learn mechanical dynamics from position data alone by leveraging the discrete forced Lagrange-d'Alembert principle. It jointly learns a conservative Lagrangian $L_ heta$ and a non-conservative force $F_ heta$ with regularization to ensure a regular, non-degenerate Lagrangian, enabling accurate long-horizon rollouts and interpretable separation of dynamics. The method is validated on synthetic tasks (damped double pendulum, dissipative charged particle) and real-world data, including autoencoder latent dynamics from pixel sequences and human motion capture, demonstrating robust predictive performance and the ability to extrapolate to conservative regimes. This work advances physics-informed learning by combining variational structure preservation with data-driven force modeling, making it applicable to real-world sensing scenarios such as motion capture and image-based tracking while enabling insights into the split between conservative and dissipative effects.

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.

Figures (9)

• Figure 1: In this paper, we propose a structure-preserving approach for learning non-conservative system that directly learns from position data only (i.e., does not require velocities or momenta).
• Figure 2: A schematic illustration of the data flow through the proposed model. We use segments of observed $(q_{n-1}, q_{n}, q_{n+1})$ to evaluate the discrete forced Euler-Lagrange equations.
• Figure 3: Combined results for Task 1 (left) and Task 2 (right). Solid Green lines are DFLNN (proposed), yellow lines are the GLNN model, and pink lines are a Neural ODE. The ground truth is indicated with dashed black. (a, c): Rollouts and extrapolation error for the trained model. (b, d): Turning off the external force component from the learned model to demonstrate the proposed model's capabilities to distinguish the conserved dynamics (only applicable for DFLNN and GLNN).
• Figure 5: Results and reconstruction (trial $2$) of human motion. Green notation is DFLNN (proposed), yellow is the GLNN model, and pink lines a Neural ODE. The ground truth is indicated in black. The left panel depicts the full movement represented as a skeletal sketch. The right panel shows the extrapolation error (top) and the rollout trajectory for the right femur (bottom).
• Figure 6: An autoencoder is applied to reduce the high-dimensional input dimension to a lower-dimensional latent space. The proposed model is applied to the dynamics in the latent space.