Discovering Symbolic Models from Deep Learning with Inductive Biases

TL;DR

This paper introduces a method to extract explicit symbolic models from deep learning, especially GNNs, by using inductive biases and symbolic regression, enabling the discovery of physical laws and improved interpretability.

Contribution

The authors propose a novel approach combining sparse representations and symbolic regression to extract explicit physical equations from trained GNNs, including new formulas in cosmology.

Findings

Successfully extracted known physical laws like force laws and Hamiltonians.

Discovered a new formula predicting dark matter concentration from simulations.

Symbolic expressions generalized better to out-of-distribution data than the original GNN.

Abstract

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.