Learning Physical Systems: Symplectification via Gauge Fixing in Dirac Structures

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.

Figures (3)

• Figure 1: (a) Our framework lifts the original phase-space to a fully conservative higher-dimensional manifold using the Dirac-lifted symplectification procedure. Then the dynamics are integrated along the surface of the lifted manifold. (b) The Dirac-lifted symplectification procedure is implemented as a flow-matching, inpainting objective using a GRU. The procedure maps the control-induced conjugate momenta ($\mathbf p_{ctrl,t}$) to the total conjugate momenta that include the system's dissipation ($\mathbf p_{0,t}$). An additional context ($T$) of 10 timesteps is supplied to the network. Then, the lifted phase-space is supplied to a SympNet that performs the next timestep prediction.
• Figure 2: (Left) Predicted conjugate momentum (green) against the actual conjugate momentum (yellow). (Right) Absolute error for the conjugate momentum.
• Figure 3: Inference Results of Dirac-Lifted dynamics prediction for the ANYmal quadruped. The 3D plot showcases the path of the quadruped's base-origin in space, with the corresponding angular momentum coordinates $p_{ang_{x}},\;p_{ang_{y}}$ and $p_{ang_{z}}$ on the right. The remainder lower-positioned plots illustrate the joint-angles and corresponding joint momenta for the Right Fore leg of the quadruped. The high overlap between the predictions (green) and the actual response (yellow), indicates the high accuracy and excellent performance of our Dirac-lifted dynamics prediction framework.