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Learning Symbolic Physics with Graph Networks

Miles D. Cranmer, Rui Xu, Peter Battaglia, Shirley Ho

TL;DR

This work tackles discovering physical laws from data by embedding physics-inspired inductive biases into graph networks. The approach enforces a linearized latent space for interactions and uses symbolic regression to extract explicit force Laws, including Newtonian gravity, from trained models. It demonstrates improved zero-shot generalization to systems with more bodies than seen during training and offers a framework to infer causal physical theories from implicit neural knowledge. Applications span n-body gravity and string-like systems, with broad potential for interpreting learned dynamics in physical sciences.

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.

Learning Symbolic Physics with Graph Networks

TL;DR

This work tackles discovering physical laws from data by embedding physics-inspired inductive biases into graph networks. The approach enforces a linearized latent space for interactions and uses symbolic regression to extract explicit force Laws, including Newtonian gravity, from trained models. It demonstrates improved zero-shot generalization to systems with more bodies than seen during training and offers a framework to infer causal physical theories from implicit neural knowledge. Applications span n-body gravity and string-like systems, with broad potential for interpreting learned dynamics in physical sciences.

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.

Paper Structure

This paper contains 4 sections, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Experiments
  4. Conclusion

Figures (3)

  • Figure 1: A schematic depicting how we extract physical knowledge from a GN.
  • Figure 2: These plots demonstrate that the graph network's messages have learned to be linear transformations of the two vector components of the true force: $f_x$ and $f_y$, for the $1/r$ law in 2D.
  • Figure 3: These plots demonstrate the improvement in generalization from minimizing the message passing space. The loss of GNs with different message-passing space dimension ($L^{e'}$), trained on a 6-body and 4-body system, in the left and right plots, respectively (indicated by the vertical line), are tested on a variable number of bodies in a $1/r^2$ simulation in 3D.