  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 relevant
variables, a task for which no general method exists. Here, we develop a
systematic approach based on the information bottleneck to identify the
relevant variables, defined as those most predictive of the future. We
elucidate analytically the relation between these relevant variables and the
eigenfunctions of the transfer operator describing the dynamics. Further, we
show that in the limit of high compression, the relevant variables are directly
determined by the slowest-decaying eigenfunctions. Our information-based
approach indicates when to optimally stop increasing the complexity of the
reduced model. Furthermore, it provides a firm foundation to construct
interpretable deep learning tools that perform model reduction. We illustrate
how these tools work in practice by considering uncurated videos of atmospheric
flows from which our algorithms automatically extract the dominant slow
collective variables, as well as experimental videos of cyanobacteria colonies
in which we discover an emergent synchronization order parameter.
