CaLiSym: Learning Symplectic Dynamics of Real-World Systems through Structured Canonical Lifts
Authors: Aristotelis Papatheodorou, Pranav Vaidhyanathan, Natalia Ares, Ioannis Havoutis, Gerard J. Milburn
Organizations: Department of Engineering Science, University of Oxford, Oxford, UK · 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
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 robotic systems of interest, where actuation, dissipation, and constraints exchange energy and momentum with the environment. We introduce CaLiSym, a lightweight framework that extends symplectic learning to such systems by changing where the geometric prior is imposed. Rather than enforcing symplecticity on the measured state, CaLiSym embeds the state and its ports into a lifted phase space, where the dynamics evolve through a symplectic map. The lift is explicit and algebraic, requiring neither recurrent latent states, transformer decoders, implicit optimization, nor inference-time numerical integration. We instantiate the framework with SympNet predictors and introduce GRB-SympNet, a B-spline variant combining approximation with exact symplectic structure. Experiments on a controlled dissipative double pendulum, a real-world quadrotor, and a contact-constrained real-world quadruped demonstrate the lowest out-of-distribution autoregressive rollout error across systems, improving by up to 69.5% while using fewer parameters and up to 85x fewer floating-point operations per step than sequence-model baselines. The lifted dynamics preserve the symplectic form to numerical precision, extending symplectic learning beyond conservative mechanics toward real-world robotics.
High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds. Although deep learning provides powerful tools for constructing low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. Based on this theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the associated symplectic projections, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method.
Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.
Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID), a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregular-step damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.