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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2506.05723 |
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| _version_ | 1866916880548102144 |
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| author | Zhou, Mo Osher, Stanley Li, Wuchen |
| author_facet | Zhou, Mo Osher, Stanley Li, Wuchen |
| contents | The Fokker-Planck (FP) equation governs the evolution of densities for stochastic dynamics of physical systems, such as the Langevin dynamics and the Lorenz system. This work simulates FP equations through a mean field control (MFC) problem. We first formulate the FP equation as a continuity equation, where the velocity field consists of the drift function and the score function, i.e., the gradient of the logarithm of the density function. Next, we design a MFC problem that matches the velocity fields in a continuity equation with the ones in the FP equation. The score functions along deterministic trajectories are computed efficiently through the score-based normalizing flow, which only rely on the derivatives of the parameterized velocity fields. A convergence analysis is conducted for our algorithm on the FP equation of Ornstein-Uhlenbeck processes. Numerical results, including Langevin dynamics, underdamped Langevin dynamics, and various chaotic systems, validate the effectiveness of our proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simulating Fokker-Planck equations via mean field control of score-based normalizing flows Zhou, Mo Osher, Stanley Li, Wuchen Optimization and Control 34K35, 37N35, 58E25, 93C15, 93C20 G.1.7 The Fokker-Planck (FP) equation governs the evolution of densities for stochastic dynamics of physical systems, such as the Langevin dynamics and the Lorenz system. This work simulates FP equations through a mean field control (MFC) problem. We first formulate the FP equation as a continuity equation, where the velocity field consists of the drift function and the score function, i.e., the gradient of the logarithm of the density function. Next, we design a MFC problem that matches the velocity fields in a continuity equation with the ones in the FP equation. The score functions along deterministic trajectories are computed efficiently through the score-based normalizing flow, which only rely on the derivatives of the parameterized velocity fields. A convergence analysis is conducted for our algorithm on the FP equation of Ornstein-Uhlenbeck processes. Numerical results, including Langevin dynamics, underdamped Langevin dynamics, and various chaotic systems, validate the effectiveness of our proposed algorithms. |
| title | Simulating Fokker-Planck equations via mean field control of score-based normalizing flows |
| topic | Optimization and Control 34K35, 37N35, 58E25, 93C15, 93C20 G.1.7 |
| url | https://arxiv.org/abs/2506.05723 |