A universal reproducing kernel Hilbert space for learning nonlinear systems operators

TL;DR

This paper introduces a universal kernel-based framework for learning nonlinear system operators, leveraging reproducing kernel Hilbert spaces to approximate complex dynamical systems with proven density and completeness properties.

Contribution

It constructs a novel class of kernel functions for nonlinear operators and proves their universality and completeness in representing system dynamics.

Findings

Kernel functions form a dense subset of nonlinear system operators.

The proposed RKHS framework is applicable to general nonlinear systems.

The approach is inspired by the universal approximation theorem for neural networks.

Abstract

In this work, we consider the problem of learning nonlinear operators that correspond to discrete-time nonlinear dynamical systems with inputs. Given an initial state and a finite input trajectory, such operators yield a finite output trajectory compatible with the system dynamics. Inspired by the universal approximation theorem of operators tailored to radial basis functions neural networks, we construct a class of kernel functions as the product of kernel functions in the space of input trajectories and initial states, respectively. We prove that for positive definite kernel functions, the resulting product reproducing kernel Hilbert space is dense and even complete in the space of nonlinear systems operators, under suitable assumptions. This provides a universal kernel-functions-based framework for learning nonlinear systems operators, which is intuitive and easy to apply to general nonlinear systems.