# Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search

arXiv: https://arxiv.org/abs/2205.13134 · HTML: https://mlsanity.com/paper/2205.13134/

## Key facts

- Cited by 66 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 55 references that are in the map.
- First posted on arXiv 26 May 2022.

## Builds on (most cited first, 16 of 16)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [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)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Symbolic Pregression: Discovering Physical Laws from Distorted Video](https://mlsanity.com/paper/2005.11212/) (2020)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [Symbolic Regression Methods for Reinforcement Learning](https://mlsanity.com/paper/1903.09688/) (2019)

## Built on by (most cited first, 30 of 66)

- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience](https://mlsanity.com/paper/2506.04282/) (2025)
- [Controllable Neural Symbolic Regression](https://mlsanity.com/paper/2304.10336/) (2023)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [LLM4ED: Large Language Models for Automatic Equation Discovery](https://mlsanity.com/paper/2405.07761/) (2024)
- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [ParFam -- (Neural Guided) Symbolic Regression Based on Continuous Global Optimization](https://mlsanity.com/paper/2310.05537/) (2023)
- [Discovering Mathematical Formulas from Data via GPT-guided Monte Carlo Tree Search](https://mlsanity.com/paper/2401.14424/) (2024)
- [Finite Expression Methods for Discovering Physical Laws from Data](https://mlsanity.com/paper/2305.08342/) (2023)
- [GENSR: Symbolic Regression Based in Equation Generative Space](https://mlsanity.com/paper/2602.20557/) (2026)
- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [RSRM: Reinforcement Symbolic Regression Machine](https://mlsanity.com/paper/2305.14656/) (2023)
- [Mimicking the Physicist's Eye:A VLM-centric Approach for Physics Formula Discovery](https://mlsanity.com/paper/2508.17380/) (2025)
- [LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search](https://mlsanity.com/paper/2606.25039/) (2026)
- [Vertical Symbolic Regression via Deep Policy Gradient](https://mlsanity.com/paper/2402.00254/) (2024)
- [Improving Monte Carlo Tree Search for Symbolic Regression](https://mlsanity.com/paper/2509.15929/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [ChatSR: Multimodal Large Language Models for Scientific Formula Discovery](https://mlsanity.com/paper/2406.05410/) (2024)
- [Racing Control Variable Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2309.07934/) (2023)
- [Complexity-Aware Deep Symbolic Regression with Robust Risk-Seeking Policy Gradients](https://mlsanity.com/paper/2406.06751/) (2024)
- [Learning Interpretable Network Dynamics via Universal Neural Symbolic Regression](https://mlsanity.com/paper/2411.06833/) (2024)
- [Bridging the Domain Gap in Equation Distillation with Reinforcement Feedback](https://mlsanity.com/paper/2505.15572/) (2025)
- [Governing Equation Discovery from Data Based on Differential Invariants](https://mlsanity.com/paper/2505.18798/) (2025)
- [Diffusion-Based Symbolic Regression](https://mlsanity.com/paper/2505.24776/) (2025)

## Abstract (arXiv)

Nonlinear dynamics is ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this fundamental issue, we propose a novel Symbolic Physics Learner (SPL) machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with state-of-the-art baselines.