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.

翻译：模型降维旨在通过少数相关变量构建对多体系统动力学的简洁而具预测性的描述。识别这些变量是模型降维的前提，然而目前尚无普适性方法完成此任务。本文基于信息瓶颈理论，提出一种识别相关变量的方法——这些变量被定义为对未来状态最具预测能力的变量。我们通过解析方法阐明了这些相关变量与描述动力学的转移算子本征函数之间的关系。在高压缩极限下，相关变量直接由衰减最慢的本征函数决定。我们的研究结果为解释自动识别降维变量的深度学习工具奠定了坚实理论基础。该方法与方程学习技术相结合，能够以数据驱动的方式揭示系统演化的隐含动力学规律。我们通过多种场景展示了这些工具的应用效果：包括可学习隐含动力学方程的理想混沌与准周期系统、未经处理的卫星大气流体运动观测数据，以及蓝藻菌群实验视频——在其中我们发现了 emergent synchronization order parameter。