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

This paper introduces a novel numerical scheme that combines a particle-based approach with a functional hierarchical tensor network ansatz and coordinate transformation to accurately solve the Fokker-Planck equation for discretized Dean-Kawasaki models in one and two dimensions.

Original authors: Xun Tang, Lexing Ying

Published 2026-02-06

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Xun Tang, Lexing Ying

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather, but instead of clouds and wind, you are tracking millions of tiny particles (like bacteria or chemicals) moving around in a box. These particles don't just move randomly; they bump into each other, get pushed by external forces, and diffuse (spread out) like ink in water.

The Dean-Kawasaki model is a complex mathematical recipe that describes exactly how the density of these particles changes over time. However, solving this recipe is incredibly hard because the number of variables is massive. If you have 64 grid cells (a modest number), you are trying to solve a puzzle with 64 dimensions simultaneously. It's like trying to find a specific spot in a room where the walls, floor, and ceiling are all made of shifting, invisible mirrors.

Here is how the authors of this paper solved that puzzle, explained through simple analogies:

1. The Problem: The "Traffic Jam" of Math

When you simulate these particles on a computer, the total amount of "stuff" (mass) must stay the same. If you have 100 units of mass, they must always add up to 100, even as they move around. Mathematically, this forces all possible states to live on a shape called a simplex (think of it as a rigid, triangular surface in high-dimensional space).

Trying to do math directly on this rigid, triangular surface is like trying to paint a picture while your hands are tied to a specific, awkward angle. It's messy and computationally expensive.

2. The First Trick: The "Unfolding" Map

To make things easier, the authors used a mathematical trick called a centered logarithmic transform.

The Analogy: Imagine the particles are stuck on a curved, rigid balloon (the simplex). To do calculations, you need to pop the balloon and lay it flat on a table (Euclidean space).
What they did: They took the particle density numbers and applied a "logarithm" (a math operation that stretches small numbers and shrinks big ones) and then centered them. This "unfolds" the rigid triangular constraint into a flat, open space where standard math tools work much better.

3. The Second Trick: The "Wavelet" Zoom Lens

Once the data is flat, the particles still have a problem: they are highly correlated. If a particle moves in one spot, its neighbors are likely to move too. This creates a "traffic jam" of information that is hard to process.

To fix this, they used a Wavelet Transform.

The Analogy: Imagine looking at a high-resolution photo of a forest. If you zoom in, you see individual leaves (fine details). If you zoom out, you see the shape of the trees (medium details). If you zoom out further, you see the whole forest (coarse details).
What they did: They broke the data down into these different "zoom levels." Instead of trying to track every single leaf and tree at once, they organized the data by scale. This revealed that the "traffic jam" disappears when you look at the data through this zoom lens. The complex correlations become simple, manageable patterns.

4. The Engine: The "Functional Hierarchical Tensor" (FHT-W)

Now that the data is unfolded and organized by zoom levels, they needed a way to store and calculate the probability of where the particles might be. A normal computer would need a library the size of a galaxy to store all the possibilities for 64 dimensions.

Instead, they used a Functional Hierarchical Tensor.

The Analogy: Think of a massive, multi-level family tree. Instead of writing down the name of every single person in the world, you only write down the relationships between parents and children. You can reconstruct the whole family tree by following the branches.
What they did: They built a "tree" structure to represent the probability density. This tree connects the different "zoom levels" (from the wavelet transform) together. By only storing the connections (the branches) rather than every single possibility, they compressed the massive amount of data into a tiny, efficient package.

5. The Result: A Clear Picture

The authors tested this method on 1D (a line) and 2D (a grid) examples with 64 grid cells.

What happened: Their method successfully predicted how the particles would behave, capturing complex interactions and correlations that other methods might miss.
The Takeaway: They proved that by "unfolding" the rigid constraints, "zooming" the data into manageable scales, and using a "tree-like" storage system, you can solve extremely complex particle physics problems that were previously too difficult for computers to handle.

In short: They took a messy, high-dimensional physics problem, flattened it, organized it by size, and compressed it into a smart tree structure, allowing them to accurately predict how crowds of particles move and interact.

Technical Summary: Solving the Fokker-Planck Equation of Discretized Dean-Kawasaki Models with Functional Hierarchical Tensor

Problem Statement
The paper addresses the challenge of solving the time-dependent Fokker-Planck equation (FPE) associated with the Dean-Kawasaki (DK) model, a stochastic partial differential equation (SPDE) describing the density fluctuations of interacting particles. While the DK model is fundamental to non-equilibrium statistical physics, fluctuating hydrodynamics, and agent-based modeling, numerically solving its associated FPE is computationally prohibitive. The primary difficulty arises from the high dimensionality of the state space: after spatial discretization into $d$ grid cells, the state is a vector of particle densities constrained to a $(d-1)$ -dimensional probability simplex (where entries are non-negative and sum to unity). Traditional grid-based methods for the FPE suffer from the curse of dimensionality, and standard density estimation techniques struggle with the simplex constraint and the strong spatial correlations inherent in the DK model.

Methodology
The authors propose a particle-based workflow that approximates the time-dependent law of the discretized DK model through density estimation, utilizing a functional hierarchical tensor network over a wavelet basis (FHT-W). The methodology proceeds through four distinct stages:

Spatial and Temporal Discretization:
The continuous DK SPDE is discretized using a finite-volume scheme on a periodic domain. This transforms the SPDE into a system of stochastic differential equations (SDEs) governing the cell-average densities $\Pi(t)$ . The discretization employs a Roe scheme for deterministic fluxes (involving external and pairwise potentials) and a specific stochastic flux formulation (based on [32]) that ensures numerical stability and preserves positivity. The resulting SDEs are simulated via Monte Carlo trajectories to generate a collection of sample paths.
Coordinate Transformation (Simplex to Euclidean Space):
To apply standard tensor network density estimation, the authors map the state from the probability simplex $\Delta$ to an unconstrained Euclidean space. This is achieved via a two-step transformation:
- Centered Log-Ratio (CLR) Transform: The density vector $\pi$ is mapped to $s$ via $s_i = \log(\pi_i) - \frac{1}{d}\sum_j \log(\pi_j)$ . This maps the interior of the simplex to a hyperplane in $\mathbb{R}^d$ where $\sum s_i = 0$ , effectively removing the simplex constraint.
- Wavelet Transform: The $s$ -coordinates are further transformed into wavelet coefficients $c$ using a discrete wavelet transform (DWT). This step exploits the multi-scale structure of the lattice model. Since adjacent grid cells in the DK model are strongly correlated due to diffusion, the wavelet transform decorrelates the variables across scales, significantly reducing the numerical rank of the resulting probability distribution $P_c(c, t)$ .
Functional Hierarchical Tensor (FHT-W) Ansatz:
The probability density in the wavelet coordinate space is approximated using a Functional Hierarchical Tensor (FHT) network. Specifically, the authors employ the FHT-W variant, which is a Hierarchical Tucker (HT) tree tensor network where the leaves correspond to wavelet coefficients.
- The tree structure mirrors the dyadic decomposition of the wavelet transform.
- The density is represented as an expansion over a fixed univariate basis (Legendre polynomials) with coefficients organized in a low-rank tensor format.
- This representation allows the storage complexity to scale linearly with the dimension $d$ rather than exponentially, provided the hierarchical ranks remain low.
Observable Estimation via Interpolation:
To compute physical observables (e.g., entropy) from the learned density, the authors propose an interpolation scheme. Since observables defined in the original $\pi$ -space become complex functions in the wavelet $c$ -space, they are compressed into the FHT-W format using a cross-interpolation method. This involves solving structured linear equations derived from point evaluations of the observable function, allowing for efficient contraction with the density tensor.

Key Contributions

A Comprehensive Computational Framework: The paper provides a complete pipeline for studying the DK model, bridging the gap between stochastic particle simulation and high-dimensional PDE solving via density estimation.
Handling Simplex Constraints: The introduction of the centered log-ratio transform combined with wavelet coordinates enables the application of tensor network methods to probability simplex problems, a domain where standard Euclidean tensor networks fail.
Scalability to High Dimensions: The method successfully handles systems with $d=64$ degrees of freedom (both 1D and 2D cases), a dimensionality that is typically intractable for direct FPE solvers or naive density estimation.
Integration of Potentials and Interactions: The framework accommodates both external potentials ( $V_1$ ) and pairwise interaction potentials ( $V_2$ ), capturing complex site-wise correlations and interaction effects.

Results
Numerical experiments in 1D and 2D demonstrate that the FHT-W approach accurately captures the time evolution of the probability law.

Accuracy: The method reproduces site-wise correlations and other statistical observables of the true model with high fidelity.
Stability: The workflow remains stable under the "strong-diffusion" regime assumption, with clamping techniques applied to the noise terms to prevent blow-up in rare events.
Efficiency: By utilizing the wavelet transform to reduce the effective rank of the distribution, the method achieves accurate results for $d=64$ with manageable tensor ranks (e.g., bond dimension $r=20$ ), avoiding the exponential storage costs associated with full tensor representations.

Significance and Claims
The authors position this work not as a novel tensor network architecture itself, but as a robust application of existing tree-based tensor networks (specifically the HT format) to a challenging class of stochastic PDEs. The primary contribution is the formulation of a practical, scalable framework for the DK model. The paper claims that this approach offers a viable alternative to traditional particle methods for studying the statistical properties of fluctuating hydrodynamics, particularly in regimes where understanding the full probability distribution (rather than just sample paths) is required. The work highlights that the combination of geometric transformations (log-ratio and wavelet) with tensor network compression is essential for overcoming the dimensionality barriers inherent in discretized interacting particle systems.