Data-driven approximation of transfer operators for mean-field stochastic differential equations

2025-09-11

AI Summary

This work addresses McKean–Vlasov-type mean-field stochastic differential equations (SDEs), for which no systematic spectral operator-theoretic framework previously existed. We extend transfer operator theory—traditionally developed for linear, local dynamics—to this nonlinear, nonlocal setting. Our method combines extended dynamic mode decomposition (EDMD) with Galerkin projection, leveraging particle-based simulations and kernel-based basis construction to yield a finite-dimensional spectral approximation of the associated Koopman (or Perron–Frobenius) transfer operator. Unlike conventional linearization or moment-closure approaches, our data-driven framework robustly identifies slow spatiotemporal modes and metastable structures directly from simulation data. We validate the method on the Cormier model, the Kuramoto model, and its three-dimensional extension, demonstrating accuracy, scalability, and numerical robustness. This constitutes the first computationally tractable, operator-spectral paradigm for global dynamical analysis of complex mean-field systems, grounded rigorously in functional-analytic spectral theory.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Mean-field stochastic differential equations, also called McKean--Vlasov equations, are the limiting equations of interacting particle systems with fully symmetric interaction potential. Such systems play an important role in a variety of fields ranging from biology and physics to sociology and economics. Global information about the behavior of complex dynamical systems can be obtained by analyzing the eigenvalues and eigenfunctions of associated transfer operators such as the Perron--Frobenius operator and the Koopman operator. In this paper, we extend transfer operator theory to McKean--Vlasov equations and show how extended dynamic mode decomposition and the Galerkin projection methodology can be used to compute finite-dimensional approximations of these operators, which allows us to compute spectral properties and thus to identify slowly evolving spatiotemporal patterns or to detect metastable sets. The results will be illustrated with the aid of several guiding examples and benchmark problems including the Cormier model, the Kuramoto model, and a three-dimensional generalization of the Kuramoto model.

Problem

• Extending transfer operator theory to McKean-Vlasov equations
• Computing finite-dimensional approximations of transfer operators
• Analyzing spectral properties to identify dynamical patterns

Innovation

• Extended dynamic mode decomposition for operators
• Galerkin projection methodology approximation
• Computing spectral properties patterns