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

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 real-world robotic systems.

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