Gaussian Process Regression for Maximum Entropy Distribution

TL;DR

This paper explores using Gaussian process regression to efficiently approximate Lagrange multipliers in maximum-entropy distributions, improving computational feasibility for moment closure problems in kinetic equations.

Contribution

It introduces a Gaussian process-based method to approximate Lagrange multipliers, addressing computational bottlenecks in maximum-entropy distribution closures.

Findings

Gaussian process regression effectively approximates Lagrange multipliers.

The method performs well on kinetic equation test cases.

Hyperparameters optimized via log-likelihood improve accuracy.

Abstract

Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations.