Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach

TL;DR

This paper employs deep neural networks to approximate solutions of the kinetic Fokker-Planck equation, analyzing their large-time behavior and boundary effects, with theoretical support for convergence to known solutions.

Contribution

It introduces a neural network approach to study the asymptotic behavior of solutions to the kinetic Fokker-Planck equation under various boundary conditions and coefficients.

Findings

Neural network solutions accurately predict large-time asymptotics.

Boundary conditions significantly influence the convergence behavior.

Theoretical proof of neural network solutions converging to analytic solutions.

Abstract

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the \textit{a priori} analytic solutions. We use the library \textit{PyTorch}, the activation function \textit{tanh} between layers, and the \textit{Adam} optimizer for the Deep Learning algorithm.