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Model reduction is the construction of simple yet predictive descriptions ofthe dynamics of many-body systems in terms of a few relevant variables. Aprerequisite to model reduction is the identification of these relevantvariables, a task for which no general method exists. Here, we develop asystematic approach based on the information bottleneck to identify therelevant variables, defined as those most predictive of the future. Weelucidate analytically the relation between these relevant variables and theeigenfunctions of the transfer operator describing the dynamics. Further, weshow that in the limit of high compression, the relevant variables are directlydetermined by the slowest-decaying eigenfunctions. Our information-basedapproach indicates when to optimally stop increasing the complexity of thereduced model. Further, it provides a firm foundation to constructinterpretable deep learning tools that perform model reduction. We illustratehow these tools work on benchmark dynamical systems and deploy them onuncurated datasets, such as satellite movies of atmospheric flows downloadeddirectly from YouTube.

Information theory for model reduction in stochastic dynamical systems