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The fruits of science are relationships made comprehensible, often by way ofapproximation. While deep learning is an extremely powerful way to findrelationships in data, its use in science has been hindered by the difficultyof understanding the learned relationships. The Information Bottleneck (IB) isan information theoretic framework for understanding a relationship between aninput and an output in terms of a trade-off between the fidelity and complexityof approximations to the relationship. Here we show that a crucial modification-- distributing bottlenecks across multiple components of the input -- opensfundamentally new avenues for interpretable deep learning in science. TheDistributed Information Bottleneck throttles the downstream complexity ofinteractions between the components of the input, deconstructing a relationshipinto meaningful approximations found through deep learning without requiringcustom-made datasets or neural network architectures. Applied to a complexsystem, the approximations illuminate aspects of the system's nature byrestricting -- and monitoring -- the information about different componentsincorporated into the approximation. We demonstrate the Distributed IB'sexplanatory utility in systems drawn from applied mathematics and condensedmatter physics. In the former, we deconstruct a Boolean circuit intoapproximations that isolate the most informative subsets of input componentswithout requiring exhaustive search. In the latter, we localize informationabout future plastic rearrangement in the static structure of a sheared glass,and find the information to be more or less diffuse depending on the system'spreparation. By way of a principled scheme of approximations, the DistributedIB brings much-needed interpretability to deep learning and enablesunprecedented analysis of information flow through a system.

The Distributed Information Bottleneck reveals the explanatory structure of complex systems