Optimizing data-driven learning of system reconstructions from incomplete low-dimensional observables*

It is frequently of interest to reconstruct the state of a high-dimensional dynamical system that is implicitly observed through an incomplete set of low-dimensional variables. Under certain technical conditions, there exist theoretical guarantees on the existence of smooth and bijective maps between embeddings of the low-dimensional observables and the high-dimensional state of the system. This presents the opportunity to reconstruct high-dimensional system states from low-dimensional observations. In previous work, we combined Takens’ Delay Embedding Theorem with numerical universal function approximators to reconstruct the atomic coordinates of molecular systems from scalar time series observations of the molecular head-to-tail distance accessible to microscopy measurements. Takens’ Theorem is, however, silent on the optimal choice and processing of low-dimensional observables. This leaves open a key practical issue: How can observables be chosen and organized to maximally embed information available about system dynamics? In this work, we demonstrate how use of multiple observable streams, incorporation of prior knowledge of system structure, and use of multiple time delays can optimize reconstruction accuracy. In a chaotic pendulum model, we demonstrate how observable choice alters reconstruction quality and how multiplexing of observable streams can improve reconstruction fidelity. In molecular simulations of a C₂₄H₅₀ polymer chain, we show how incorporating prior knowledge alters reconstruction performance and we relate these choices to dynamical properties such Lyapunov exponents. In a Lotka–Volterra model of predator-prey dynamics, we show how dynamics evolving on multiple time scales are best reconstructed when multiplexing observable time delays on multiple time scales. We extract actionable guidelines from these studies that can be used to improve reconstruction and prediction quality for arbitrary dynamical systems in diverse fields including epidemiology, climatology, and econometrics.