mlsanityData to 7 Oct 2026

Learning Symbolic Physics with Graph Networks

arXiv 1909.05862 ↗First posted 12 Sep 2019cs.LG astro-ph.IM physics.comp-phcited by 52

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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.

Key facts. Cited by 52 papers in the map (data through 7 Oct 2026). First posted on arXiv 12 Sep 2019.

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Citing papers per year: 2021: 17; 2022: 9; 2023: 8; 2024: 8; 2025: 9; 2026: 1.

52 papers.

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  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2021Graph Neural Networks with Learnable Structural and Positional Representationscited by 326
  3. 2022A Generalization of ViT/MLP-Mixer to Graphscited by 93
  4. 2021Beltrami Flow and Neural Diffusion on Graphscited by 84
  5. 2022Rediscovering orbital mechanics with machine learningcited by 71
  6. 2021Entropy-based Logic Explanations of Neural Networkscited by 51
  7. 2021Logic Explained Networkscited by 45
  8. 2023Interpretable Machine Learning for Science with PySR and SymbolicRegression.jlcited by 36
  9. 2025A Survey of Scientific Large Language Models: From Data Foundations to Agent Frontierscited by 34
  10. 2023The transformative potential of machine learning for experiments in fluid mechanicscited by 26
  11. 2021An Approach to Symbolic Regression Using Feyncited by 20
  12. 2023Machine Learning for Partial Differential Equationscited by 20
38 more
  1. 2021Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Futurecited by 18
  2. 2021A Bayesian neural network predicts the dissolution of compact planetary systemscited by 15
  3. 2024Symmetry-Informed Governing Equation Discoverycited by 14
  4. 2021Applying Machine Learning to Study Fluid Mechanicscited by 13
  5. 2021Symbolic regression outperforms other models for small data setscited by 12
  6. 2021Maximum Entropy Weighted Independent Set Pooling for Graph Neural Networkscited by 12
  7. 2022SYMBA: Symbolic Computation of Squared Amplitudes in High Energy Physics with Machine Learningcited by 12
  8. 2025Interpretable Machine Learning in Physics: A Reviewcited by 11
  9. 2021Identifying Physical Law of Hamiltonian Systems via Meta-Learningcited by 8
  10. 2022Simulating Liquids with Graph Networkscited by 8
  11. 2022Is the Machine Smarter than the Theorist: Deriving Formulas for Particle Kinematics with Symbolic Regressioncited by 8
  12. 2021Unsupervised Resource Allocation with Graph Neural Networkscited by 7
  13. 2023Categorical Foundations of Explainable AI: A Unifying Theorycited by 7
  14. 2022Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scattercited by 6
  15. 2022Learning Mechanically Driven Emergent Behavior with Message Passing Neural Networkscited by 6
  16. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  17. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  18. 2021Deep Learning of Conjugate Mappingscited by 5
  19. 2023Oracle-Preserving Latent Flowscited by 5
  20. 2024Symbolic Regression for Beyond the Standard Model Physicscited by 5
  21. 2022textttMangrove: Learning Galaxy Properties from Merger Treescited by 4
  22. 2024Exploring the Truth and Beauty of Theory Landscapes with Machine Learningcited by 4
  23. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  24. 2021Interpretable Embedding Procedure Knowledge Transfer via Stacked Principal Component Analysis and Graph Neural Networkcited by 3
  25. 2024Graph External Attention Enhanced Transformercited by 3
  26. 2023Seeking Truth and Beauty in Flavor Physics with Machine Learningcited by 2
  27. 2021Graph convolutional network for predicting abnormal grain growth in Monte Carlo simulations of microstructural evolutioncited by 1
  28. 2022AI Research Associate for Early-Stage Scientific Discoverycited by 1
  29. 2023Interpretable Neural PDE Solvers using Symbolic Frameworkscited by 1
  30. 2023Deep learning complete intersection Calabi-Yau manifoldscited by 1
  31. 2024Multi-Hierarchical Surrogate Learning for Structural Dynamical Crash Simulations Using Graph Convolutional Neural Networkscited by 1
  32. 2024Data-driven Modeling in Metrology -- A Short Introduction, Current Developments and Future Perspectivescited by 1
  33. 2024Decomposing heterogeneous dynamical systems with graph neural networkscited by 1
  34. 2024Explaining Hypergraph Neural Networks: From Local Explanations to Global Conceptscited by 1
  35. 2025Positional Encoding meets Persistent Homology on Graphscited by 1
  36. 2025From Embeddings to Equations: Genetic-Programming Surrogates for Interpretable Transformer Classificationcited by 1
  37. 2025Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physicscited by 1
  38. 2021Improving Simulations with Symmetry Control Neural Networkscited by 0

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