Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structures
This addresses a foundational limitation in physics-informed deep learning for constrained, dissipative systems like legged robots, enabling more stable and accurate geometric models.
The paper tackled the problem of learning physical systems with dissipation and constraints, which degrade symplectic geometry and stability, by introducing Presymplectification Networks (PSNs) to restore non-degenerate symplectic structures via Dirac structures, demonstrating it on the ANYmal quadruped robot for improved constrained trajectory forecasting.
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.