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.

翻译：本文提出了一种基于函数张量网络拟设的数值格式，用于求解离散化Dean-Kawasaki模型的Fokker-Planck方程。Dean-Kawasaki模型描述了相互作用粒子系统的密度涨落，是一个高度奇异的随机偏微分方程。通过对Dean-Kawasaki模型进行有限体积离散化，我们导出了一个随机微分方程。为完整刻画离散化Dean-Kawasaki模型，我们求解了该随机微分方程动力学所对应的Fokker-Planck方程。特别地，我们采用基于粒子的方法，通过从模拟轨迹中执行一系列密度估计任务来获得Fokker-Planck方程的解，并利用函数层次张量模型来表征密度。针对样本轨迹支撑集位于单纯形上的挑战，我们通过对数参数化将坐标从单纯形变换到欧几里得空间，随后对变换后的变量执行基于素描技术的密度估计流程。该方法具有通用性，可应用于单纯形上的一般密度估计任务。我们将所提方法成功应用于一维和二维Dean-Kawasaki模型，并证明该方法在存在外势场和粒子相互作用的情况下仍能保持高精度。