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

arXiv: https://arxiv.org/abs/2509.09891 · HTML: https://mlsanity.com/paper/2509.09891/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 59 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Builds on (most cited first, 8 of 8)

- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Consensus Convergence with Stochastic Effects](https://mlsanity.com/paper/1508.07313/) (2015)
- [Computation and Control of Unstable Steady States for Mean Field Multiagent Systems](https://mlsanity.com/paper/2406.11725/) (2024)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)

## Abstract (arXiv)

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.