The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker-Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.

翻译：Fokker-Planck方程描述了随机微分方程中概率密度的演化过程。随着系统维度的增加，使用传统数值方法求解这一偏微分方程（PDE）的计算成本变得难以承受。本文提出一种快速、可扩展且可解释的高维Fokker-Planck方程求解方法。该方法将近似解表示为具有时变均值与协方差的形状变形高斯函数的线性组合，并通过降阶非线性解方法（RONS）控制这些参数的演化，确保近似解始终逼近PDE的真实解。因此，所提方法不仅能近似瞬态动力学，还能在平衡密度存在时逼近该平衡密度。我们的近似解可视为嵌入概率密度空间中的有限维统计流形上的演化过程。我们证明了该流形上的RONS度量张量与Fisher信息矩阵等价。同时，我们将该方法解释为具有高斯激活函数和时变参数的浅层神经网络。与现有深度学习方法相比，本方法具有可解释性、无需训练，并自动确保近似解满足概率密度的所有性质。