Solving the Fokker–Planck Equation of Discretized Dean–Kawasaki Models with Functional Hierarchical Tensor

Abstract.

We propose a particle-based workflow for approximating the time-dependent law of finite-volume discretizations of the Dean–Kawasaki model. After discretization, the state is a nonnegative vector whose total mass is conserved by the finite-volume scheme, and it is therefore supported on a probability simplex. To enable tensor-network density estimation, we map the simplex to an unconstrained Euclidean space using a centered logarithmic transform and then apply a wavelet transform that organizes degrees of freedom by spatial scale. On the transformed variables, we fit the probability density with a functional hierarchical tensor over a wavelet basis, i.e., a hierarchical-Tucker/tree-tensor-network representation of the coefficient tensor of a fixed univariate basis expansion. We illustrate the method on 1D and 2D examples with [Formula: see text] degrees of freedom, including cases with external potentials and pairwise interactions. The method accurately captures the site-wise correlations and other observables of the true model.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Xun-Tang123/FHT_for_deans_equation and in the supplementary materials ( FHT_for_deans_equation-main.zip [27.5MB]). [Formula: see text]