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Discovering Symbolic Models from Deep Learning with Inductive Biases
Miles Cranmer · Alvaro Sanchez Gonzalez · Peter Battaglia · Rui Xu · Kyle Cranmer · David Spergel · Shirley Ho

Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1591

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.

Author Information

Miles Cranmer (Princeton University)
Alvaro Sanchez Gonzalez (DeepMind)
Peter Battaglia (DeepMind)
Rui Xu (Princeton University)
Kyle Cranmer (New York University)

Kyle Cranmer is an Associate Professor of Physics at New York University and affiliated with NYU's Center for Data Science. He is an experimental particle physicists working, primarily, on the Large Hadron Collider, based in Geneva, Switzerland. He was awarded the Presidential Early Career Award for Science and Engineering in 2007 and the National Science Foundation's Career Award in 2009. Professor Cranmer developed a framework that enables collaborative statistical modeling, which was used extensively for the discovery of the Higgs boson in July, 2012. His current interests are at the intersection of physics and machine learning and include inference in the context of intractable likelihoods, development of machine learning models imbued with physics knowledge, adversarial training for robustness to systematic uncertainty, the use of generative models in the physical sciences, and integration of reproducible workflows in the inference pipeline.

David Spergel (Flatiron Institute)
Shirley Ho (Flatiron institute/ New York University/ Carnegie Mellon)

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