Koopman Spectral Analysis and System Identification for Stochastic Dynamical Systems via Yosida Approximation of Generators

TL;DR

This work introduces a resolvent-Yosida based framework for learning the generator of stochastic dynamical systems from data, enabling Koopman spectral analysis and robust system identification under limited and noisy observations. By combining finite-horizon stopped-process resolvent, finite-rank projections, and martingale-problem-based convergence, the authors develop RT-EDMD, a data-driven algorithm that estimates the Koopman generator and its spectrum from Euler–Maruyama samples. The method achieves accurate identification of drift and diffusion terms and reliable extraction of dominant spectral modes, demonstrated on Ornstein–Uhlenbeck and noisy Lotka–Volterra systems, outperforming EDMD and gEDMD, especially at low sampling rates or from single trajectories. Theoretical guarantees via martingale arguments and weak convergence underpin the approach, while practical convergence analyses ensure reliability when replacing expectations with empirical averages. Overall, the framework offers a robust, theoretically grounded tool for stochastic system identification and spectral analysis with broad applicability in physics, biology, and finance.

Abstract

System identification and Koopman spectral analysis are crucial for uncovering physical laws and understanding the long-term behaviour of stochastic dynamical systems governed by stochastic differential equations (SDEs). In this work, we propose a novel method for estimating the Koopman generator of systems of SDEs, based on the theory of resolvent operators and the Yosida approximation. This enables both spectral analysis and accurate estimation and reconstruction of system parameters. The proposed approach relies on only mild assumptions about the system and effectively avoids the error amplification typically associated with direct numerical differentiation. It remains robust even under low sampling rates or with only a single observed trajectory, reliably extracting dominant spectral modes and dynamic features. We validate our method on two simple systems and compare it with existing techniques as benchmarks. The experimental results demonstrate the effectiveness and improved performance of our approach in system parameter estimation, spectral mode extraction, and overall robustness.

Figures (2)

• Figure 1: Spectral estimation error of the Ornstein–Uhlenbeck (OU) process at different frequencies
• Figure 2: Reconstruction of sample paths from the OU process

Theorems & Definitions (1)

• proof