Fisher information and shape-morphing modes for solving the Fokker-Planck equation in higher dimensions

TL;DR

This paper introduces a scalable, interpretable method for solving the high-dimensional Fokker-Planck equation using shape-morphing Gaussians and Fisher information, avoiding deep learning complexities.

Contribution

The authors develop a novel approach that approximates solutions with shape-morphing Gaussians, linking Fisher information to the solution dynamics, and ensuring interpretability and probabilistic properties.

Findings

Method effectively approximates transient and equilibrium densities in high dimensions.

Fisher information matrix coincides with the RONS metric tensor, providing theoretical insight.

Approach is interpretable, no training required, and maintains probability density properties.

Abstract

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.