Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structures

TL;DR

This paper introduces Presymplectification Networks (PSNs), a novel framework that learns to restore symplectic geometry in constrained, dissipative systems, enabling structure-preserving modeling of complex robotic dynamics.

Contribution

The work presents the first method to learn symplectification lifts via Dirac structures, extending geometric deep learning to constrained and dissipative systems.

Findings

Successfully applied to the ANYmal quadruped robot dynamics.

Restores symplectic invariants in constrained, dissipative systems.

Enables energy and momentum preservation in learned models.

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.