CALISYM Learning Symplectic Dynamics of Real-World Systems through Structured Canonical Lifts

Jul 16, 2026

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Physics-informed learning enables data-efficient, stable dynamics prediction, yet its strongest geometric guarantees typically apply to closed conservative systems, leaving many robotic scenarios uncovered where actuation, dissipation, and constraints continuously exchange energy and momentum with the environment. CALISYM introduces a lightweight framework that extends exact symplectic learning by imposing the geometric prior on a structured lifted canonical phase space, using an explicit algebraic lift. Learned dynamics evolve via exactly symplectic maps, avoiding recurrent latent states, transformer decoders, implicit optimization, or inference-time ODE integration. Experiments show improved out-of-distribution autoregressive prediction and symplectic form preservation to numerical precision.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages18

Reading Time45 min

Introduction

Dynamics modeling in robotics

Limits of classical model-based control

Limits of current learned dynamics in non-conservative regimes

CALISYM approach overview

Lifted canonical phase space and gauge-fixed embedding

Exactly symplectic map and projection back to physical state

CALISYM: Learning Symplectic Dynamics of Real-World Systems through Structured Canonical Lifts Aristotelis Papatheodorou∗ , 1 Pranav Vaidhyanathan∗ , 1 Natalia Ares 1 Ioannis Havoutis 1 Gerard J. Milburn2 arXiv :2607 .06824v 1 [ cs .RO] 7 Jul 2026 Abstract—Physics-informed learning promises data-efficient and stable dynamics prediction, yet its strongest geometric guarantees have largely remained confined to closed conservative systems. This excludes many robotic systems of practical interest, where actuation, dissipation, and constraints continuously exchange energy and momentum with the environment. We introduce CALISYM†, a lightweight framework that extends exact symplectic learning to such systems by changing where the geometric prior is imposed. Rather than enforcing symplecticity on the measured physical state, CALISYM embeds the state and its physical ports into a structured lifted canonical phase space, where the learned dynamics evolve through an exactly symplectic map. The lift is explicit and algebraic, requiring neither recurrent latent states, transformer decoders, implicit optimization, nor inference-time ODE integration. We instantiate the framework with generalized-ridge SYMPNET predictors and introduce GRB-SYMPNET, a B-spline variant that combines local approximation with exact symplectic structure. Experiments on a controlled dissipative double pendulum, a real-world quadrotor, and a contact-rich quadruped demonstrate consistent improvements in out-of-distribution autoregressive prediction while using parameter-efficient models. At the same time, the learned lifted dynamics preserve the symplectic form to numerical precision. These results show that symplectic learning can be extended beyond conservative mechanics through structured canonical lifts, enabling geometry-preserving dynamics models for realworld robotic systems. Index Terms—robot dynamics learning, symplectic neural networks, geometric machine learning, structure-preserving learning, Hamiltonian systems, contact-rich robotics, model-based control. I. INTRODUCTION DYNAMICS has been studied extensively, producing a wide range of methods for explaining, predicting, and controlling the physical systems around us. The arrival of computation extended that reach to systems of scale and complexity that no closed-form analysis could touch. Amongst other fields, this shift has been highly consequential for robotics, where accurate dynamics models now underpin nearly every aspect of estimation, prediction and control. However, accuracy almost always comes with a cost. 1Aristotelis Papatheodorou, Pranav Vaidhyanathan, Natalia Ares and Ioannis Havoutis are with the Department of Engineering Science, University of Oxford, Oxford, [UK. A.P. is](UK. A.P. is) supported by Oxford’s Clarendon Fund and the JPMorgan Chase AI PhD [Fellowship. P.V. is](Fellowship. P.V. is) supported by the United States Army Research Office under Award No. W911NF-21-S-0009-2 . N.A. acknowledges support from the European Research Council (Grant agreement 948932) and the Royal Society (URF-R1-191150) . 2 School of Mathematics and Physics, University of Sussex, Brighton, BN1 9RH, UK. National Centre for Quantum Computing, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX UK ∗ Equal Contribution {aristotelis, [pranav](pranav}@robots.ox.ac.uk)[}](pranav}@robots.ox.ac.uk)[@robots.ox.ac.uk](pranav}@robots.ox.ac.uk)[ ](pranav}@robots.ox.ac.uk)†Open-source implementation will be released upon acceptance. Fig. 1. Overview of CALISYM. The measured state lives in the physical phase space T∗ Q, the cotangent bundle of the configuration manifold Q. Because the system exchanges energy and impulses through actuation, dissipation, and contact, the physical dynamics on T∗ Q need not be closed or symplectic. CALISYM therefore embeds the physical state xt = (qt , pt) and its current ports et into a gauge-fixed data section Set of the lifted phase space T

FAQs

What problem does CALISYM address in learned symplectic dynamics?

CALISYM addresses the gap that most geometric/symplectic learning guarantees are limited to closed conservative systems, while real robots involve actuation, dissipation, and constraints that exchange energy and momentum with the environment.

How does CALISYM enforce symplecticity in systems with non-conservative effects?

CALISYM embeds the physical state and its physical ports into a structured lifted canonical phase space, then advances the lifted state using an exactly symplectic map and projects back to the physical state.

What does CALISYM avoid in its model design and rollout procedure?

The framework does not require recurrent latent states, transformer decoders, implicit optimization, or inference-time ODE integration; the lift is explicit and algebraic, with symplectic evolution handled by the learned map.

Jul 16, 2026

Upload by Skyler