Learning Symbolic Physics with Graph Networks

Abstract

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.

6 pages; references added + improvements to writing and clarity; accepted for an oral presentation at Machine Learning and the Physical Sciences Workshop @ NeurIPS 2019

References in corpus (5)

• Neural Message Passing for Quantum Chemistry
• Geometric deep learning: going beyond Euclidean data
• Interaction Networks for Learning about Objects, Relations and Physics
• Graph networks as learnable physics engines for inference and control
• Neural Relational Inference for Interacting Systems

Cited by in corpus (23)

• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
• A Generalization of Transformer Networks to Graphs
• The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence
• Discovering Symbolic Models from Deep Learning with Inductive Biases
• Benchmarking Graph Neural Networks
• Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
• Logic Explained Networks
• Entropy-based Logic Explanations of Neural Networks
• Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future
• Symbolic regression outperforms other models for small data sets
• Graph Neural Networks with Learnable Structural and Positional Representations
• Back to the Formula -- LHC Edition
• Modeling assembly bias with machine learning and symbolic regression
• Beltrami Flow and Neural Diffusion on Graphs
• Deep Learning of Conjugate Mappings
• Maximum Entropy Weighted Independent Set Pooling for Graph Neural Networks
• Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data
• Unsupervised Resource Allocation with Graph Neural Networks
• Inferring halo masses with Graph Neural Networks
• Identifying Physical Law of Hamiltonian Systems via Meta-Learning
• Uncertainty in Neural Relational Inference Trajectory Reconstruction
• Improving Simulations with Symmetry Control Neural Networks
• An Experimental Study of the Transferability of Spectral Graph Networks