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

2025-05-26

AI Summary

This work addresses the problem of learning equations of motion for mechanical systems solely from discrete positional observations—such as motion-capture trajectories, pixel coordinates, or low-resolution tracking data—without requiring explicit velocity measurements. We propose a neural dynamics modeling framework grounded in the discrete forced Lagrange–d’Alembert principle, the first to integrate this variational principle with feedforward neural networks and autoencoder latent spaces: conservative and non-conservative forces are learned separately, while symplectic geometric structure is inherently preserved. Physical consistency is enforced by constraining the model to satisfy the discrete Euler–Lagrange equations. The method is validated on synthetic mechanical systems, real human motion capture data, and latent embeddings of image sequences, achieving high-fidelity trajectory reconstruction and interpretable decomposition into conservative and external forcing terms. Our core contribution is the first purely position-driven dynamical learning paradigm that simultaneously ensures physical fidelity, data efficiency, and structural interpretability.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

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.

Problem

• Learning mechanical systems from position data without velocity measurements
• Decomposing dynamics into conservative and non-conservative components
• Validating method on synthetic and real-world datasets like human motion

Innovation

• Uses discrete Lagrange-d'Alembert principle for dynamics
• Learns conservative and non-conservative components separately
• Applies neural networks to system identification tasks