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. Knowing the state of a dynamical system as it evolves in time is important for a variety of applications. This paper proposes a general-purpose method for inferring unmeasured state variables from a limited set of ongoing measurements. Our method is intended for situations in which mathematical models of system dynamics are unavailable or are insufficiently accurate to perform the desired inference. We use the machine-learning technique called “reservoir computing,” with which we construct a system-independent means of processing the measurements. A key point is the extent to which this approach is “universal.” That is, our examples show that the same reservoir can be trained to infer the state of different systems. It is the training that relates to a specific system, not the “hardware.” The reservoir hardware plays a similar role to an animal's brain, which retrains itself as the system represented by its body and environment changes.