Discovering Symbolic Models from Deep Learning with Inductive Biases

Author(s): Miles Cranmer, Alvaro Sanchez-Gonzalez, Peter Battaglia, Rui Xu, Kyle Cranmer, David Spergel, Shirley Ho
Venue: arXiv
Year: 2020

Paper: https://arxiv.org/abs/2006.11287

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.

Additional information

Video review of the paper: https://www.youtube.com/watch?v=LMb5tvW-UoQ

Blog: https://astroautomata.com/paper/symbolic-neural-nets/

Github: https://github.com/MilesCranmer/symbolic_deep_learning

Twitter thread: https://twitter.com/MilesCranmer/status/1275241300829384704