Solving the Fokker-Planck equation of discretized Dean-Kawasaki models with functional hierarchical tensor

TL;DR

This work addresses solving the high-dimensional Fokker-Planck equation arising from discretized Dean-Kawasaki models by combining a particle-based density estimation workflow with a wavelet-based, tree-structured functional tensor network (FHT-W) representation. It transforms the simplex-constrained state to Euclidean coordinates via a two-step mapping $s(\\pi)$ and a subsequent wavelet transform to obtain $c$, enabling efficient density estimation on a lower-rank tensor manifold. The authors develop an interpolation framework for observable functions within the FTN, using sketched linear systems and SVD to compute tensor cores, and demonstrate accurate density reconstructions and observable estimates in 1D and 2D DK models, both with and without external potentials. Overall, the method provides a scalable, particle-based approach to high-dimensional simplex density estimation with controlled errors, suitable for DK-type SPDEs and related lattice-density problems.

Abstract

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.

Figures (5)

• Figure 1: Illustration of the FHT-W ansatz for $L = 4$. Each $v_{k, l}$ represents the tensor component $G_{v_{k, l}}$ at the external node $v_{k, l}$, and the $v_{k, l}$ corresponds to the variable $c_{k, l}$ in the wavelet transformation. Each $w_{k, l}$ represents the tensor component $G_{w_{k, l}}$ at the internal node $w_{k, l}$.
• Figure 1: 1D Dean-Kawasaki model with $V_1 = V_2 = 0$. Plot of the correlation matrix predicted by the FHT-W ansatz.
• Figure 2: (A) A tree structure $T = (V, E)$ with $V = V_{\mathrm{ext}}= \{1, \ldots, 10\}$. (B) Tensor Diagram representation of a tree tensor network over $T$.
• Figure 3: Interpolations for 1D Dean-Kawasaki model and 2D Dean-Kawasaki model without potential functions. The target function is $M(c) =M(s^{-1}(\mathrm{IDWT}(c))$, where $M(\pi)$ is the Shannon entropy function. The plot compares the function evaluation between $M(c)$ and the FHT-W ansatz obtained from the interpolation procedure in \ref{['alg:tree-based FTN interpolation internal']}. The $20000$ evaluation points are chosen randomly. The mean relative prediction error is $8.7\times 10^{-9}$ for 1D Dean-Kawasaki and $1.7\times 10^{-6}$ for 2D Dean-Kawasaki.
• Figure 4: 2D Dean-Kawasaki model with $V_1 = V_2 = 0$. The plot of the two-point correlation function predicted by the FHT-W ansatz.

Theorems & Definitions (1)

• Definition 3.1