Solving the Fokker-Planck equation of discretized Dean-Kawasaki models with functional hierarchical tensor
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914306994470912 |
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| author | Tang, Xun Ying, Lexing |
| author_facet | Tang, Xun Ying, Lexing |
| contents | We introduce a novel numerical scheme for solving the Fokker-Planck equation of discretized Dean-Kawasaki models with a functional tensor network ansatz. The Dean-Kawasaki model describes density fluctuations of interacting particle systems, and it is a highly singular stochastic partial differential equation. By performing a finite-volume discretization of the Dean-Kawasaki model, we derive a stochastic differential equation (SDE). To fully characterize the discretized Dean-Kawasaki model, we solve the associated Fokker-Planck equation of the SDE dynamics. In particular, we use a particle-based approach whereby the solution to the Fokker-Planck equation is obtained by performing a series of density estimation tasks from the simulated trajectories, and we use a functional hierarchical tensor model to represent the density. To address the challenge that the sample trajectories are supported on a simplex, we apply a coordinate transformation from the simplex to a Euclidean space by logarithmic parameterization, after which we apply a sketching-based density estimation procedure on the transformed variables. Our approach is general and can be applied to general density estimation tasks over a simplex. We apply the proposed method successfully to the 1D and 2D Dean-Kawasaki models. Moreover, we show that the proposed approach is highly accurate in the presence of external potential and particle interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solving the Fokker-Planck equation of discretized Dean-Kawasaki models with functional hierarchical tensor Tang, Xun Ying, Lexing Numerical Analysis We introduce a novel numerical scheme for solving the Fokker-Planck equation of discretized Dean-Kawasaki models with a functional tensor network ansatz. The Dean-Kawasaki model describes density fluctuations of interacting particle systems, and it is a highly singular stochastic partial differential equation. By performing a finite-volume discretization of the Dean-Kawasaki model, we derive a stochastic differential equation (SDE). To fully characterize the discretized Dean-Kawasaki model, we solve the associated Fokker-Planck equation of the SDE dynamics. In particular, we use a particle-based approach whereby the solution to the Fokker-Planck equation is obtained by performing a series of density estimation tasks from the simulated trajectories, and we use a functional hierarchical tensor model to represent the density. To address the challenge that the sample trajectories are supported on a simplex, we apply a coordinate transformation from the simplex to a Euclidean space by logarithmic parameterization, after which we apply a sketching-based density estimation procedure on the transformed variables. Our approach is general and can be applied to general density estimation tasks over a simplex. We apply the proposed method successfully to the 1D and 2D Dean-Kawasaki models. Moreover, we show that the proposed approach is highly accurate in the presence of external potential and particle interaction. |
| title | Solving the Fokker-Planck equation of discretized Dean-Kawasaki models with functional hierarchical tensor |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.22816 |