Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structures

2025-06-23

AI Summary

Traditional symplectic learning methods fail for dissipative, holonomically constrained multibody systems (e.g., legged robots) due to degeneracy of the phase-space presymplectic structure. Method: We propose Presymplectification Networks—a geometric deep learning framework that lifts the degenerate presymplectic manifold to a higher-dimensional non-degenerate symplectic manifold via Dirac structures; it employs a recurrent encoder and flow matching for end-to-end dynamics modeling, and—novelly in symplectic deep learning—incorporates gauge-fixing principles from constrained mechanics to construct a learnable symmetry-preserving architecture. Contribution/Results: Integrating SympNet, geometric constraint modeling, and conservation-law priors, our method achieves significantly improved long-horizon trajectory prediction accuracy and stability in contact-rich scenarios on the ANYmal platform, thereby bridging a critical theoretical and practical gap in geometric machine learning for dissipative constrained systems.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Physics-informed deep learning has achieved remarkable progress by embedding geometric priors, such as Hamiltonian symmetries and variational principles, into neural networks, enabling structure-preserving models that extrapolate with high accuracy. However, in systems with dissipation and holonomic constraints, ubiquitous in legged locomotion and multibody robotics, the canonical symplectic form becomes degenerate, undermining the very invariants that guarantee stability and long-term prediction. In this work, we tackle this foundational limitation by introducing Presymplectification Networks (PSNs), the first framework to learn the symplectification lift via Dirac structures, restoring a non-degenerate symplectic geometry by embedding constrained systems into a higher-dimensional manifold. Our architecture combines a recurrent encoder with a flow-matching objective to learn the augmented phase-space dynamics end-to-end. We then attach a lightweight Symplectic Network (SympNet) to forecast constrained trajectories while preserving energy, momentum, and constraint satisfaction. We demonstrate our method on the dynamics of the ANYmal quadruped robot, a challenging contact-rich, multibody system. To the best of our knowledge, this is the first framework that effectively bridges the gap between constrained, dissipative mechanical systems and symplectic learning, unlocking a whole new class of geometric machine learning models, grounded in first principles yet adaptable from data.

Problem

• Degenerate symplectic form in dissipative constrained systems
• Learning symplectification via Dirac structures for stability
• Bridging constrained systems with symplectic geometric learning

Innovation

• Introduces Presymplectification Networks (PSNs) for symplectification
• Combines recurrent encoder with flow-matching objective
• Uses lightweight Symplectic Network (SympNet) for constrained trajectories