Sroczynski David W, Dietrich Felix, Koronaki Eleni D, Talmon Ronen, Coifman Ronald R, Bollt Erik, Kevrekidis Ioannis G
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA.
School of Computation, Information and Technology, Technical University of Munich, 80333 Munich, Germany.
PNAS Nexus. 2024 Dec 6;3(12):pgae494. doi: 10.1093/pnasnexus/pgae494. eCollection 2024 Dec.
Before we attempt to (approximately) learn a function between two sets of observables of a physical process, we must first decide what the and of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of first deciding "the right quantities" to relate through such a function, and then proceeding to learn it. This is accomplished by first processing simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: records of multiple of the system. We determine (i) what subsets of observables are between the observation processes (and therefore observable from each other, relatable through a function); and (ii) what information is to these common observables, therefore particular to each observation process, and not contributing to the desired function. Any data-driven technique can subsequently be used to learn the input-output relation-from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular "twists" of the approach are discussed. The first has to do with the of particular quantities of interest from the measurements. We now construct mappings from set of observations from one process to of measurements of the second process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures "now," while the second observation process measures "in the future," the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.
在我们尝试（近似地）学习物理过程的两组可观测量之间的函数之前，我们必须首先确定所需函数的输入和输出是什么。在这里，我们展示了两种不同的、数据驱动的方法，首先确定通过这样一个函数相关联的“正确量”，然后继续学习它。这是通过首先处理来自物理系统观测的同步异构数据流（时间序列集合）来实现的：系统多个属性的记录。我们确定（i）观测过程之间哪些可观量子集是相关的（因此彼此可观测，可通过函数关联）；以及（ii）哪些信息对于这些共同的可观测量是特定的，因此特定于每个观测过程，并且对所需函数没有贡献。随后可以使用任何数据驱动技术来学习输入 - 输出关系——从k近邻和几何调和函数到高斯过程和神经网络。讨论了该方法的两个特殊“变体”。第一个与从测量中提取特定感兴趣量有关。我们现在构建从一个过程的一组观测到第二个过程的测量集的映射，与这一组一致。第二个试图将我们的框架与一种因果关系形式联系起来：如果一个观测过程测量“现在”，而第二个观测过程测量“未来”，那么在观测过程中共同的部分之间要学习的函数构成了系统演化的动态模型。
