Model reduction is the construction of simple yet predictive descriptions of the dynamics of many-body systems in terms of a few relevant variables. A prerequisite to model reduction is the identification of these variables, a task for which no general method exists. Here, we develop an approach to identify relevant variables, defined as those most predictive of the future, using the so-called information bottleneck. We elucidate analytically the relation between these relevant variables and the eigenfunctions of the transfer operator describing the dynamics. In the limit of high compression, the relevant variables are directly determined by the slowest-decaying eigenfunctions. Our results provide a firm foundation to interpret deep learning tools that automatically identify reduced variables. Combined with equation learning methods this procedure yields the hidden dynamical rules governing the system's evolution in a data-driven manner. We illustrate how these tools work in diverse settings including model chaotic and quasiperiodic systems in which we also learn the underlying dynamical equations, uncurated satellite recordings of atmospheric fluid flows, and experimental videos of cyanobacteria colonies in which we discover an emergent synchronization order parameter.

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