The Fokker-Planck (FP) equation is a foundational PDE in stochastic processes. However, curse of dimensionality (CoD) poses challenge when dealing with high-dimensional FP PDEs. Although Monte Carlo and vanilla Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimension increases, surpassing the precision of numerical simulations and resulting in substantial errors. Moreover, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a difficult HJB equation, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in SDEs. The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling. Three fitting methods, Score Matching (SM), Sliced SM (SSM), and Score-PINN, are introduced. The proposed score-based SDE solver operates in two stages: first, employing SM, SSM, or Score-PINN to acquire the score; and second, solving the LL via an ODE using the obtained score. Comparative evaluations across these methods showcase varying trade-offs. The proposed method is evaluated across diverse SDEs, including anisotropic OU processes, geometric Brownian, and Brownian with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy. The numerical results demonstrate the score-based SDE solver's stability, speed, and performance across different settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

翻译：福克-普朗克（FP）方程是随机过程中的基础偏微分方程。然而，在处理高维FP偏微分方程时，“维数灾难”（CoD）带来了严峻挑战。尽管蒙特卡洛方法和原始物理信息神经网络（PINNs）已展现出应对CoD的潜力，但两种方法在处理与布朗运动相关的概率密度函数（PDF）时，均会在高维情形下产生数值误差。随着维数增加，逐点PDF值呈指数级递减，超出数值模拟精度范围，导致显著误差。此外，蒙特卡洛方法因需大量采样，难以实现快速采样。通过原始PINNs对对数似然（LL）进行建模，会将FP方程转化为求解困难的HJB方程，其误差随维数增长而急剧扩大。为此，我们提出一种基于得分（score）的新型求解器，用于拟合随机微分方程（SDEs）中的得分函数。得分函数定义为LL的梯度，在推断LL和PDF中起基础作用，并能实现快速SDE采样。本文引入了三种拟合方法：得分匹配（SM）、切片得分匹配（SSM）和Score-PINN。所提出的基于得分的SDE求解器分两阶段运行：首先采用SM、SSM或Score-PINN获取得分；然后利用所得得分通过常微分方程（ODE）求解LL。不同方法的对比评估展示了各自的性能权衡。该方法在多类SDE（包括各向异性Ornstein-Uhlenbeck过程、几何布朗运动及变特征空间布朗运动）上进行了评估，并测试了多种分布（高斯分布、对数正态分布、拉普拉斯分布和柯西分布）。数值结果表明，基于得分的SDE求解器在不同场景下均具备稳定性、高效性和优异性能，巩固了其作为高维FP方程CoD解决方案的潜力。