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

arXiv: https://arxiv.org/abs/2607.06824 · HTML: https://mlsanity.com/paper/2607.06824/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 79 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 30 of 37)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Recurrent World Models Facilitate Policy Evolution](https://mlsanity.com/paper/1809.01999/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Self Forcing: Bridging the Train-Test Gap in Autoregressive Video Diffusion](https://mlsanity.com/paper/2506.08009/) (2025)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [TD-MPC2: Scalable, Robust World Models for Continuous Control](https://mlsanity.com/paper/2310.16828/) (2023)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Professor Forcing: A New Algorithm for Training Recurrent Networks](https://mlsanity.com/paper/1610.09038/) (2016)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Robotic World Model: A Neural Network Simulator for Robust Policy Optimization in Robotics](https://mlsanity.com/paper/2501.10100/) (2025)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Multi-Step Prediction of Dynamic Systems with Recurrent Neural Networks](https://mlsanity.com/paper/1806.00526/) (2018)
- [Learning-based legged locomotion; state of the art and future perspectives](https://mlsanity.com/paper/2406.01152/) (2024)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Physics-informed Neural Networks to Model and Control Robots: a Theoretical and Experimental Investigation](https://mlsanity.com/paper/2305.05375/) (2023)
- [A Survey on Imitation Learning for Contact-Rich Tasks in Robotics](https://mlsanity.com/paper/2506.13498/) (2025)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Model Predictive Control for Micro Aerial Vehicles: A Survey](https://mlsanity.com/paper/2011.11104/) (2020)
- [Agile Maneuvers in Legged Robots: a Predictive Control Approach](https://mlsanity.com/paper/2203.07554/) (2022)
- [GrandTour: A Legged Robotics Dataset in the Wild for Multi-Modal Perception and State Estimation](https://mlsanity.com/paper/2602.18164/) (2026)

## Abstract (arXiv)

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.