# A universal reproducing kernel Hilbert space for learning nonlinear systems operators

arXiv: https://arxiv.org/abs/2412.18360 · HTML: https://mlsanity.com/paper/2412.18360/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2024.

## Built on by (most cited first, 1 of 1)

- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)

## Abstract (arXiv)

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.