Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions
Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions
复制标题
用于求解高维福克普朗克方程的费希尔信息和形状变形模式
DOI：
10.1016/j.amc.2023.128489
复制
发表时间：
2024
影响因子：
4
通讯作者：
Farazmand, Mohammad
中科院分区：
文献类型：
--
作者：
Anderson, William;Farazmand, Mohammad
福克-普朗克方程描述了与随机微分方程相关的概率密度的演化。随着系统维数的增加，使用传统的数值方法求解偏微分方程（PDE）在计算上变得非常困难。在这里，我们介绍了一种快速，可扩展，可解释的方法来解决福克-普朗克方程，适用于更高的维度。该方法将解近似为形状变形高斯与时间相关均值和协方差的线性组合。这些参数根据降阶非线性解决方案（RONS）的方法进行演化，该方法确保近似解始终接近PDE的真实解。因此，所提出的方法近似的瞬态动力学以及平衡密度，当后者存在。我们的近似解可以看作是一个有限维的概率密度空间中嵌入的统计流形上的演变。我们证明了RONS中的度量张量与该流形上的Fisher信息矩阵相一致。我们还讨论了解释我们的方法作为一个浅神经网络与高斯激活函数和时变参数。与现有的深度学习方法相比，我们的方法是可解释的，不需要训练，并自动确保近似解满足概率密度的所有属性。
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.