On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them

TL;DR

The paper tackles the challenge of learning relations between heterogeneous observations of a shared dynamical system by first identifying which observables are common across sensors and which are sensor-specific. It introduces two data-driven frameworks—Alternating Diffusion Maps and Jointly Smooth Functions—to extract common coordinates and then learns cross-sensor observer functions using KNN, Geometric Harmonics, and FFNN. It also develops notions of causality via present-to-future mappings and extends to mixed (dirty) channels with Output-Informed Diffusion Maps to parameterize level sets consistent with other sensors. Through three nonlinear dynamical examples (periodic, quasiperiodic, chaotic) and multiple sensor configurations, the work demonstrates how common dynamics can be isolated and modeled, enabling cross-sensor learning and generating pathways toward causal and multiscale modeling. The findings have implications for sensor fusion, data-driven causal modeling, and multiscale closure modeling in complex dynamical systems.

Abstract

Before we attempt to 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 initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: 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 of one process to entire level sets of measurements of the 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.

Figures (21)

• Figure 1: Illustrative Sensor setup: Sensor 1 only observes parts of systems X and Y. Sensor 2 only observes parts of systems X and Z.
• Figure 2: Results of running LLR on the set of successive alternating-diffusion eigenvectors $\phi_i$ (sorted by eigenvalue). $\phi_1$ is trivially constant, and $\phi_2$ has a normalized LLR residual of 1 by definition. $\phi_2$ is the only other top eigenvector with a high residual, indicating that it represents a unique direction and that the most parsimonious embedding of the common system is two-dimensional.
• Figure 3: These plots confirm that the alternating-diffusion embedding is one-to-one/bi-Lipschitz with the coordinates of the common system X. (top) Plots of the alternating-diffusion embedding colored by $\theta_A^{(X)}$ (left) and $\theta_B^{(X)}$ (right). (bottom) Plots of $\theta_B^{(X)}$ vs. $\theta_A^{(X)}$, colored by alternating-diffusion eigenvectors 2 (left) and 3 (right).
• Figure 4: Plots of the alternating-diffusion embedding colored by each of the individual sensor channels, with the LLR residual above each plot. Channels 1--4 of Sensor 1 (top row) are the measurements $\lbrack\theta_{A}^{(X)}\left( t \right),\theta_{A}^{(Y)}\left( t \right),\theta_{A}^{(X)}\left( t - \Delta t \right),\theta_{A}^{(Y)}\left( t - \Delta t \right)\rbrack$, while channels 1--4 of Sensor 2 (bottom row) are the measurements $\lbrack\theta_{B}^{(X)}\left( t \right),y\left( t \right),\theta_{B}^{(X)}\left( t - \Delta t \right),\ y\left( t - \Delta t \right)\rbrack$. Coordinates that belong to the common system (Sensor 1 channels 1 and 3, Sensor 2 channels 1 and 3) have a low residual and appear visually smooth. Other coordinates have a high residual and appear noisy.
• Figure 5: (a) The first 10 extracted jointly smooth functions. (b)(Left)The embedding result for the two most parsimonious JSFs. (Right) The original system X data colored by one JSF.