Solution of the Fokker-Planck equation by cross approximation method in the tensor train format

TL;DR

This paper introduces a new numerical scheme combining Chebyshev interpolation, spectral differentiation, and tensor train approximations to efficiently solve high-dimensional Fokker-Planck equations, reducing computational complexity.

Contribution

It presents a novel tensor train-based method with cross approximation for solving multidimensional Fokker-Planck equations, enhancing efficiency and scalability.

Findings

Effective in multidimensional problems like Ornstein-Uhlenbeck process

Significant reduction in degrees of freedom needed

Applicable to density estimation in machine learning

Abstract

We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.