Learning Symbolic Equations with Deep Learning

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.

Shirley Ho

Shirley Ho is Acting Director of the Center for Computational Astrophysics (CCA) at the Flatiron Institute, where she leads the Cosmology X Data Science Group and works with a wonderful group of international collaborators. She is also a research professor at New York University and an associate (adjunct) professor in the Department of Physics at Carnegie Mellon University. Ho's research interests have ranged from fundamental cosmological measurements to exoplanet statistics to using machine learning to estimate how much dark matter is in the universe. She has broad expertise in theoretical astrophysics, observational astronomy, and data science. Ho’s recent focus has been on understanding and developing novel tools in deep learning techniques, and applying them to astrophysical challenges. Her goal is to understand the universe’s beginning, evolution, and ultimate fate.

Since 2011, she has been a primary mentor to more than 15 postdoctoral fellows, 6 graduate students, and 14 undergraduates in the fields of astrophysics, computer science, and statistics. She plans to continue mentoring future generations of astrophysicists and data scientists at CCA and other institutions. Her recognitions include the NASA Group Achievement Award, Macronix Prize, Carnegie Science Award, and election as International Astrostatistics Association Fellow. Ho also participated in the first ever ACM-IMS Interdisciplinary Summit on the Foundations of Data Science in 2019