Learning Nonlinear Systems via Volterra Series and Hilbert-Schmidt Operators

This article examines the application of regularization techniques and kernel methods in addressing the task of learning nonlinear dynamical systems from input-output data. Our assumption is that the estimator belongs to the space of polynomials composed of Hilbert-Schmidt operators, which ensures the ability to approximate nonlinear dynamics arbitrarily, even within bounded but noncompact data domains. By employing regularization techniques, we propose a finite-dimensional identification procedure that exhibits computational complexity proportional to the square of the size of the training set size. This procedure is applicable to a broad range of systems, including discrete and continuous time nonlinear systems on finite or infinite dimensional state spaces. We delve into the selection of the regularization parameter, taking into account the measurement noise, and also discuss the incorporation of causality constraints. Furthermore, we explore how to derive estimates of the Volterra series of the operator by selecting a parametric inner product between data trajectories.