# Learning Symbolic Physics with Graph Networks

arXiv: https://arxiv.org/abs/1909.05862 · HTML: https://mlsanity.com/paper/1909.05862/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2019.

## Built on by (most cited first, 30 of 53)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [A Generalization of ViT/MLP-Mixer to Graphs](https://mlsanity.com/paper/2212.13350/) (2022)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Entropy-based Logic Explanations of Neural Networks](https://mlsanity.com/paper/2106.06804/) (2021)
- [Logic Explained Networks](https://mlsanity.com/paper/2108.05149/) (2021)
- [A Survey of Scientific Large Language Models: From Data Foundations to Agent Frontiers](https://mlsanity.com/paper/2508.21148/) (2025)
- [The transformative potential of machine learning for experiments in fluid mechanics](https://mlsanity.com/paper/2303.15832/) (2023)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [An Approach to Symbolic Regression Using Feyn](https://mlsanity.com/paper/2104.05417/) (2021)
- [Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future](https://mlsanity.com/paper/2108.04543/) (2021)
- [A Bayesian neural network predicts the dissolution of compact planetary systems](https://mlsanity.com/paper/2101.04117/) (2021)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Applying Machine Learning to Study Fluid Mechanics](https://mlsanity.com/paper/2110.02083/) (2021)
- [SYMBA: Symbolic Computation of Squared Amplitudes in High Energy Physics with Machine Learning](https://mlsanity.com/paper/2206.08901/) (2022)
- [Symbolic regression outperforms other models for small data sets](https://mlsanity.com/paper/2103.15147/) (2021)
- [Maximum Entropy Weighted Independent Set Pooling for Graph Neural Networks](https://mlsanity.com/paper/2107.01410/) (2021)
- [Identifying Physical Law of Hamiltonian Systems via Meta-Learning](https://mlsanity.com/paper/2102.11544/) (2021)
- [Simulating Liquids with Graph Networks](https://mlsanity.com/paper/2203.07895/) (2022)
- [Is the Machine Smarter than the Theorist: Deriving Formulas for Particle Kinematics with Symbolic Regression](https://mlsanity.com/paper/2211.08420/) (2022)
- [Unsupervised Resource Allocation with Graph Neural Networks](https://mlsanity.com/paper/2106.09761/) (2021)
- [Categorical Foundations of Explainable AI: A Unifying Theory](https://mlsanity.com/paper/2304.14094/) (2023)
- [Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter](https://mlsanity.com/paper/2201.01305/) (2022)
- [Learning Mechanically Driven Emergent Behavior with Message Passing Neural Networks](https://mlsanity.com/paper/2202.01380/) (2022)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)

## Abstract (arXiv)

We introduce an approach for imposing physically motivated inductive biases on graph networks to learn interpretable representations and improved zero-shot generalization. Our experiments show that our graph network models, which implement this inductive bias, can learn message representations equivalent to the true force vector when trained on n-body gravitational and spring-like simulations. We use symbolic regression to fit explicit algebraic equations to our trained model's message function and recover the symbolic form of Newton's law of gravitation without prior knowledge. We also show that our model generalizes better at inference time to systems with more bodies than had been experienced during training. Our approach is extensible, in principle, to any unknown interaction law learned by a graph network, and offers a valuable technique for interpreting and inferring explicit causal theories about the world from implicit knowledge captured by deep learning.