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. Further, it provides a firm foundation to construct interpretable deep learning tools that perform model reduction. We illustrate how these tools work on benchmark dynamical systems and deploy them on uncurated datasets, such as satellite movies of atmospheric flows downloaded directly from YouTube.

翻译：模型降阶是通过少量相关变量构建多体系统动力学的简洁且具有预测性的描述。实现模型降阶的前提是识别这些相关变量，而目前尚无通用方法完成该任务。本文基于信息瓶颈理论，提出了一种系统化方法来识别相关变量——定义为对未来状态最具预测性的变量。我们从解析角度阐明了这些相关变量与描述动力学的传递算子特征函数之间的关系。进一步证明，在高压缩极限下，相关变量直接由最慢衰减的特征函数决定。基于信息论的方法能够指示何时应最优地停止增加降阶模型的复杂度，同时为构建可解释的深度学习工具（用于执行模型降阶）奠定了坚实基础。我们通过基准动力系统演示了这些工具的工作原理，并将其应用于未整理数据集（如直接从YouTube下载的大气流动卫星影像）。