O N L EARNING WHAT TO L EARN : HETEROGENEOUS OBSERVATIONS OF DYNAMICS AND ESTABLISHING ( POSSIBLY CAUSAL ) RELATIONS AMONG THEM APREPRINT David W. Sroczynski Department of Chemical and Biological Engineering Princeton University Princeton NJ, USA Felix Dietrich School of Computation, Information and Technology Technical University of Munich Munich, Germany Eleni D. Koronaki Faculty of Science, Technology and Medicine University of Luxembourg Esch-sur-Alzette, Luxembourg Ronen Talmon Viterbi Faculty of Electrical Engineering, Technion Israel Institute of Technology Haifa, Israel Ronald R. Coifman School of Engineering & Applied Science Yale University New Haven CT, USA Erik Bollt Electrical & Computer Engineering Clarkson University Potsdam NY, USA Ioannis G. Kevrekidis ∗ Department of Chemical and Biomolecular Engineering Department of Applied Mathematics and Statistics Department of Urology Johns Hopkins University Baltimore MD, USA June 12, 2024 ABSTRACT Before we attempt to (approximately) learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs 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 multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: records of multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation 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 identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations from one process to entire level sets of measurements of the second process, consistent with this single set. The second attempts to relate our framework to a form of ∗ Correspondong author: yannisk@jhu.edu arXiv:2406.06812v1 [cs.LG] 10 Jun 2024