Koopman Identification of Nonlinear Systems via Reservoir Liftings

TL;DR

This paper introduces RC-Koopman, a reservoir computing-based framework for learning linear representations of nonlinear systems, improving stability and accuracy over existing methods by controlling reservoir memory.

Contribution

It proposes a novel RC-Koopman approach that interprets reservoirs as finite-dimensional Koopman dictionaries with spectral radius tuning for better system identification.

Findings

RC-Koopman achieves better reconstruction accuracy than EDMD and Hankel methods.

The spectral radius selection aligns reservoir memory with system timescales.

The framework ensures well-posedness and numerical stability via the Echo State Property.

Abstract

Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC) paradigm, this paper introduces the RC-Koopman framework, which interprets reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is explicitly controlled by its spectral radius. We show that the Echo State Property (ESP) guarantees well-posedness and favorable numerical conditioning of the lifted Koopman approximation. A correlation-based spectral radius selection algorithm aligns reservoir memory with dominant system timescales. Analysis reveals how the finite memory of the reservoir determines which Koopman eigenfunctions remain observable from the lifted features. Evaluation on synthetic benchmarks demonstrates that RC-Koopman achieves a favorable balance between reconstruction accuracy of the underlying nonlinear dynamics and dynamical stability, compared to Extended Dynamic Mode Decomposition (EDMD) and Hankel-based lifting approaches. Code available at: https://github.com/NEAR-the-future/RC-Koopman.git