Reservoir Computing with an Inertial Form

Abstract

We propose a framework of a reservoir with an inertial manifold, which is a finite-dimensional object that attracts all trajectories. In addition, based on the theoretical results concerning infinite-dimensional dynamical systems, we demonstrate that this reservoir should have some desirable properties called the echo state property and the separation property. By considering the projection of the system onto the inertial manifold, we describe the behavior of the reservoir as a system of finite-dimensional ordinary differential equations called an inertial form. Next, we present a finite-dimensional model that approximates the inertial form. Finally, we propose a temporally discretized model, which is more suitable for computer computation. We give the theoretical proofs of our theorems herein.