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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2509.12841 |
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| _version_ | 1866916952057839616 |
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| author | Cai, Zhiqiang Liu, Chengyu Zhou, Xiang |
| author_facet | Cai, Zhiqiang Liu, Chengyu Zhou, Xiang |
| contents | Stochastic interacting particle systems are widely used to model collective phenomena across diverse fields, including statistical physics, biology, and social dynamics. The McKean-Vlasov equation arises as the mean-field limit of such systems as the number of particles tends to infinity, while its long-time behaviour is characterized by stationary distributions as time tends to infinity. However, the validity of interchanging the infinite-time and infinite-particle limits is not guaranteed. Consequently, simulation methods that rely on a finite-particle truncation may fail to accurately capture the mean-field system's stationary distributions, particularly when the coexistence of multiple metastable states leads to phase transitions. In this paper, we adapt the framework of the Weak Generative Sampler (WGS) -- a generative technique based on normalizing flows and a weak formulation of the nonlinear Fokker-Planck equation -- to compute and generate i.i.d. samples satisfying the stationary distributions of McKean-Vlasov processes. Extensive numerical experiments validate the efficacy of the proposed methods, showcasing their ability to accurately approximate stationary distributions and capture phase transitions in complex systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak Generative Sampler for Stationary Distributions of McKean-Vlasov System Cai, Zhiqiang Liu, Chengyu Zhou, Xiang Computational Physics Stochastic interacting particle systems are widely used to model collective phenomena across diverse fields, including statistical physics, biology, and social dynamics. The McKean-Vlasov equation arises as the mean-field limit of such systems as the number of particles tends to infinity, while its long-time behaviour is characterized by stationary distributions as time tends to infinity. However, the validity of interchanging the infinite-time and infinite-particle limits is not guaranteed. Consequently, simulation methods that rely on a finite-particle truncation may fail to accurately capture the mean-field system's stationary distributions, particularly when the coexistence of multiple metastable states leads to phase transitions. In this paper, we adapt the framework of the Weak Generative Sampler (WGS) -- a generative technique based on normalizing flows and a weak formulation of the nonlinear Fokker-Planck equation -- to compute and generate i.i.d. samples satisfying the stationary distributions of McKean-Vlasov processes. Extensive numerical experiments validate the efficacy of the proposed methods, showcasing their ability to accurately approximate stationary distributions and capture phase transitions in complex systems. |
| title | Weak Generative Sampler for Stationary Distributions of McKean-Vlasov System |
| topic | Computational Physics |
| url | https://arxiv.org/abs/2509.12841 |