# Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps

arXiv: https://arxiv.org/abs/2103.05632 · HTML: https://mlsanity.com/paper/2103.05632/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 9 Mar 2021.

## Builds on (most cited first, 15 of 15)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data](https://mlsanity.com/paper/1905.10396/) (2019)
- [Tensorized Transformer for Dynamical Systems Modeling](https://mlsanity.com/paper/2006.03445/) (2020)

## Built on by (most cited first, 30 of 41)

- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [SAMformer: Unlocking the Potential of Transformers in Time Series Forecasting with Sharpness-Aware Minimization and Channel-Wise Attention](https://mlsanity.com/paper/2402.10198/) (2024)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Context parroting: A simple but tough-to-beat baseline for foundation models in scientific machine learning](https://mlsanity.com/paper/2505.11349/) (2025)
- [Locally-symplectic neural networks for learning volume-preserving dynamics](https://mlsanity.com/paper/2109.09151/) (2021)
- [Learning Large-Time-Step Molecular Dynamics with Graph Neural Networks](https://mlsanity.com/paper/2111.15176/) (2021)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Analysing Multi-Task Regression via Random Matrix Theory with Application to Time Series Forecasting](https://mlsanity.com/paper/2406.10327/) (2024)
- [Hamiltonian Neural Networks with Automatic Symmetry Detection](https://mlsanity.com/paper/2301.07928/) (2023)
- [Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets](https://mlsanity.com/paper/2303.17824/) (2023)
- [Symmetry Preservation in Hamiltonian Systems: Simulation and Learning](https://mlsanity.com/paper/2308.16331/) (2023)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Recurrences reveal shared causal drivers of complex time series](https://mlsanity.com/paper/2301.13516/) (2023)
- [Approximation capabilities of measure-preserving neural networks](https://mlsanity.com/paper/2106.10911/) (2021)
- [Learning Trajectories of Hamiltonian Systems with Neural Networks](https://mlsanity.com/paper/2204.05077/) (2022)
- [ModLaNets: Learning Generalisable Dynamics via Modularity and Physical Inductive Bias](https://mlsanity.com/paper/2206.12325/) (2022)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models](https://mlsanity.com/paper/2209.07646/) (2022)
- [CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2408.16160/) (2024)
- [Post-Training in Time Series Foundation Models: A Unifying Framework](https://mlsanity.com/paper/2607.20002/) (2026)
- [VPNets: Volume-preserving neural networks for learning source-free dynamics](https://mlsanity.com/paper/2204.13843/) (2022)
- [A hybrid approach for solving the gravitational N-body problem with Artificial Neural Networks](https://mlsanity.com/paper/2310.20398/) (2023)
- [Learning the action for long-time-step simulations of molecular dynamics](https://mlsanity.com/paper/2508.01068/) (2025)
- [Discovering Phase Space Structure in Learned Hamiltonian Systems](https://mlsanity.com/paper/2509.23003/) (2025)

## Abstract (arXiv)

We consider the learning and prediction of nonlinear time series generated by a latent symplectic map. A special case is (not necessarily separable) Hamiltonian systems, whose solution flows give such symplectic maps. For this special case, both generic approaches based on learning the vector field of the latent ODE and specialized approaches based on learning the Hamiltonian that generates the vector field exist. Our method, however, is different as it does not rely on the vector field nor assume its existence; instead, it directly learns the symplectic evolution map in discrete time. Moreover, we do so by representing the symplectic map via a generating function, which we approximate by a neural network (hence the name GFNN). This way, our approximation of the evolution map is always exactly symplectic. This additional geometric structure allows the local prediction error at each step to accumulate in a controlled fashion, and we will prove, under reasonable assumptions, that the global prediction error grows at most linearly with long prediction time, which significantly improves an otherwise exponential growth. In addition, as a map-based and thus purely data-driven method, GFNN avoids two additional sources of inaccuracies common in vector-field based approaches, namely the error in approximating the vector field by finite difference of the data, and the error in numerical integration of the vector field for making predictions. Numerical experiments further demonstrate our claims.