Discovering Symbolic Models from Deep Learning with Inductive Biases (2006.11287v2)

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.

• The paper introduces a novel framework that trains GNNs with sparse latent representations and employs symbolic regression to uncover underlying physical laws.
• It applies regularization techniques like L1 and KL divergence to constrain message-passing layers, successfully rediscovering classical force laws and energy functions in Newtonian and Hamiltonian systems.
• The method generalizes well to unseen data, as shown in a cosmological study where the extracted symbolic expressions improved dark matter halo predictions over traditional estimators.

Extracting Symbolic Representations from Graph Neural Networks

The paper presents a novel methodology for extracting symbolic representations from deep learning models, specifically focusing on Graph Neural Networks (GNNs). This approach enables the interpretation of neural networks by imposing strong inductive biases, thereby allowing the extraction of symbolic expressions that reveal the physical relationships encoded within the models.

Methodology Overview

The authors propose a framework where GNNs are trained with a focus on sparse latent representations. This is achieved by using regularization techniques like L$_1$ regularization and Kullback-Leibler (KL) divergence to constrain the message-passing layer of the GNN, which reflects known physical interactions. Symbolic regression is then employed to distill these learned latent representations into explicit symbolic expressions that describe the underlying physical laws.

Experimental Framework

The framework is validated across multiple case studies including Newtonian dynamics, Hamiltonian systems, and a complex cosmological dataset:

1. Newtonian Dynamics: The method is used to rediscover known force laws from simulations of interacting particles. By constraining the dimensionality of the message passing in GNNs to match the physical dimensionality of the force vectors, the authors successfully extract symbolic expressions corresponding to classical forces, such as spring and gravitational forces.
2. Hamiltonian Dynamics: A modified GNN structure, termed Flattened Hamiltonian Graph Network (FlatHGN), is utilized to learn energy functions rather than force vectors. The paper demonstrates that symbolic regression can capture pairwise potential energy relations from the extracted representations, offering new insights into Hamiltonian dynamics.
3. Cosmological Study: In a novel application, the authors apply their method to a cosmological dataset to predict dark matter halo overdensities based on surrounding structures. The extracted symbolic formula shows improved predictive accuracy over manually designed estimators, highlighting the method's potential for discovering new physical insights in complex astrophysical systems.

Implications and Future Work

The results showcase the potential for combining the power of high-dimensional deep learning models with the interpretability of symbolic mathematics. The extracted symbolic representations often generalized better on out-of-distribution data than the original GNNs, suggesting a paradigm shift towards integrating symbolic reasoning in neural networks.

This approach opens up new avenues for AI, where interpretable models can be automatically generated from data. Such models can be pivotal in scientific domains where understanding the underlying physics is as important as making accurate predictions. Future research may expand on integrating more complex symmetries into GNNs, exploring alternative neural architectures, or further developing symbolic regression tools to enhance the interpretability and utility of AI models in scientific inquiry.

Overall, the paper introduces a compelling argument for leveraging traditional symbolic reasoning in conjunction with modern deep learning, providing a means to not only improve the interpretability of neural networks but also potentially drive scientific discoveries.