Discovering Symbolic Models from Deep Learning with Inductive Biases
This work addresses the interpretability of neural networks and discovery of physical principles, offering a method for extracting symbolic knowledge from learned representations, though it is incremental as it builds on existing techniques like symbolic regression.
The paper tackled the problem of distilling symbolic models from deep learning by introducing inductive biases, specifically in Graph Neural Networks (GNNs), and successfully extracted known physical equations and a new analytic formula for dark matter concentration from simulations, with the symbolic expressions generalizing better than the GNN itself.
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.