Reservoir observers: Model-free inference of unmeasured variables in chaotic systems

Deducing the state of a dynamical system as a function of time from a limited number of concurrent system state measurements is an important problem of great practical utility. A scheme that accomplishes this is called an "observer." We consider the case in which a model of the system is unavailable or insufficiently accurate, but "training" time series data of the desired state variables are available for a short period of time, and a limited number of other system variables are continually measured. We propose a solution to this problem using networks of neuron-like units known as "reservoir computers." The measurements that are continually available are input to the network, which is trained with the limited-time data to output estimates of the desired state variables. We demonstrate our method, which we call a "reservoir observer," using the Rössler system, the Lorenz system, and the spatiotemporally chaotic Kuramoto-Sivashinsky equation. Subject to the condition of observability (i.e., whether it is in principle possible, by any means, to infer the desired unmeasured variables from the measured variables), we show that the reservoir observer can be a very effective and versatile tool for robustly reconstructing unmeasured dynamical system variables.