An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis

TL;DR

The paper tackles the challenge of computing the invariant density for randomly perturbed dynamical systems by solving the stationary Fokker-Planck equation using a data-driven hybrid solver that couples Monte Carlo references with a projection onto $\mathrm{Ker}(\mathbf{A})$. It then introduces a block solver to divide the computational domain into small blocks, enabling scalable, parallelizable computation for low- to moderate-dimensional problems, and proposes two techniques—overlapping blocks and shifting blocks—to reduce interface errors between blocks. The authors provide a convergence analysis and demonstrate, through ring, Rossler, and MMO examples, that the approach achieves accurate invariant densities with substantially lower cost than global solvers while preserving important dynamical structures. This framework broadens the practical applicability of data-driven Fokker-Planck solvers to higher resolution, localized domains, and multi-dimensional systems, with potential extensions to time-dependent problems and mesh-free implementations.

Abstract

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker-Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker-Planck equations proposed in [15]. Based on these results, we propose a new `block solver' for the stationary Fokker-Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

Figures (15)

• Figure 1: Diagonal entries from the $R$ matrix of the $QR$ decomposition of the basis $\mathcal{B}$ in \ref{['laplace_basis']} and \ref{['laplace_basis_2']} respectively arranged in lexocographical order. The basis functions are nearly orthogonal. The mesh size $N$ is $101$. Value of last diagonal entry is $0.015$.
• Figure 2: Principal angles between $\mathrm{Ker}(\mathbf{A})$ and $\Theta_{D}$ for $D = 1, 2, 3$. Left: Diffusion process without drift. Right: Fokker-Planck equation as given in Example \ref{['Ring density function']}.
• Figure 3: Empirical spatial distribution of error term for the ring density function as in Section \ref{['Ring density function']}. Top left: $\mathbf{v} - \mathbf{u}^{\text{ext}}$. Top right: $\mathbf{u} - \mathbf{u}^{\text{ext}}$. Middle left: $\mathbf{v}' - \mathbf{u}^{\text{ext}}$. Middle right: $\mathbf{u}' - \mathbf{u}^{\text{ext}}$. Bottom: Comparison of $\rho_{v}$, $\rho_{u}$, $\rho_{v'}$, $\rho_{u'}$ for $D = 1, 2, 3, 4$.
• Figure 4: Left: Some trajectories of the deterministic part of equation \ref{['ring']}. Right: Exact solution of the Fokker-Planck equation for \ref{['ring']}.
• Figure 5: (Ring density) The approximation by Monte Carlo simulation (left) and the algorithm in Section \ref{['Algorithm description']} (right) with $256\times256$ mesh points and $10^7$ samples.

Theorems & Definitions (3)

• Theorem 2.1
• proof
• Proposition 2.2