# Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems

arXiv: https://arxiv.org/abs/2209.01521 · HTML: https://mlsanity.com/paper/2209.01521/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 40 references that are in the map.
- First posted on arXiv 4 Sep 2022.

## Builds on (most cited first, 21 of 21)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Residual Attention Network for Image Classification](https://mlsanity.com/paper/1704.06904/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Sparse Symplectically Integrated Neural Networks](https://mlsanity.com/paper/2006.12972/) (2020)

## Built on by (most cited first, 1 of 1)

- [H-FEX: A Symbolic Learning Method for Hamiltonian Systems](https://mlsanity.com/paper/2506.20607/) (2025)

## Abstract (arXiv)

Here we present Symplectically Integrated Symbolic Regression (SISR), a novel technique for learning physical governing equations from data. SISR employs a deep symbolic regression approach, using a multi-layer LSTM-RNN with mutation to probabilistically sample Hamiltonian symbolic expressions. Using symplectic neural networks, we develop a model-agnostic approach for extracting meaningful physical priors from the data that can be imposed on-the-fly into the RNN output, limiting its search space. Hamiltonians generated by the RNN are optimized and assessed using a fourth-order symplectic integration scheme; prediction performance is used to train the LSTM-RNN to generate increasingly better functions via a risk-seeking policy gradients approach. Employing these techniques, we extract correct governing equations from oscillator, pendulum, two-body, and three-body gravitational systems with noisy and extremely small datasets.