Gaussian Process Regression for Maximum Entropy Distribution
This work addresses a specific computational challenge in moment closure for kinetic equations, but it is incremental as it applies an existing method (Gaussian processes) to a known bottleneck.
The paper tackled the computational bottleneck of finding Lagrange multipliers for maximum-entropy distributions in moment closure problems by using Gaussian process regression to approximate these multipliers as a function of moments, achieving performance tested on non-equilibrium distributions governed by kinetic equations.
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.