(560e) What Matters and What Does Not Matter: Parametrizing Common and Sensor-Specific Information across Multiple Sensors in Chemically Reacting Systems

We consider the case where two or more sensors measure information from a common reaction system of interest, but each sensor's measurements also contain information from independent sensor-specific dynamics ("the disjoint'', e.g. sensor-specific noise). In this context, manifold learning methods (i.e., Diffusion Maps [1]) can be modified to isolate the components of the measurements that correspond to the common system. We compare the results of Alternating Diffusion Maps [2] and Jointly Smooth Functions [3] in parametrizing the common system. Depending on the scenario, we can either (a) find which sensor variables can be learned as functions of the other sensor's variables (i.e. what nonlinear observers we can construct) or (b), given a "common" measurement that matters, find all possible instances of what does not matter consistent with it (an appropriate level set). In certain cases we can even learn evolution equations for the common system.

From this point, it is desirable to also parametrize each uncommon system. We demonstrate an approach using Output-Influenced Diffusion Maps [4], as well as a more reliable approach using Neural Networks to find parametrizations of the disjoint features; these are (in the original sensor data) locally conformal to the common system parametrization.

References:
[1] R.R. Coifman and S. Lafon, Appl. Comput. Harmon. Anal. 21, 5 (2006).
[2] R.R. Lederman and R. Talmon, Appl. Comput. Harmon. Anal. 44, 509 (2018).
[3] O. Yair, F. Dietrich, R. Mulayoff, R. Talmon, and I.G. Kevrekidis, Spectral discovery of jointly smooth features for multimodal data, ArXiv (2020).
[4] A. Holiday, M. Kooshkbaghi, J.M. Bello-Rivas, C. William Gear, A. Zagaris, and I.G. Kevrekidis, J. Comput. Phys. 392, 419 (2019).