# Gaussian Process Regression for Maximum Entropy Distribution

**Authors:** Mohsen Sadr, Manuel Torrilhon, M. Hossein Gorji  
**arXiv:** [2308.06149](https://arxiv.org/abs/2308.06149)  
**Date:** 2023-08-11  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2308.06149v1/

## Key points

- Gaussian processes used to approximate Lagrange multipliers in Maximum-Entropy Distributions
- Lagrange multipliers computationally challenging to find in moment closure problems
- Investigating if Gaussian priors can effectively approximate Lagrange multipliers based on given moments
- Examining different kernel functions and optimizing hyperparameters through maximizing log-likelihood
- Evaluating performance of data-driven Maximum-Entropy closure approach using Bhatnagar-Gross-Krook and Boltzmann kinetic equations
- Novel method for addressing moment closure problems by leveraging Gaussian processes
- Promising performance demonstrated in various test cases, potential for practical closure settings

## Summary

The paper titled "Gaussian Process Regression for Maximum Entropy Distribution" explores the use of Gaussian processes to approximate the Lagrange multipliers in Maximum-Entropy Distributions. These distributions are commonly used in moment closure problems, where finding the appropriate Lagrange multipliers can be computationally challenging. Motivated by the success of Gaussian processes in other applications, the authors investigate whether Gaussian priors can effectively approximate the Lagrange multipliers based on a given set of moments. To do this, they examine various kernel functions and optimize hyperparameters by maximizing the log-likelihood. The performance of this data-driven Maximum-Entropy closure approach is then evaluated using several test cases. Specifically, the relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations is analyzed. Overall, this research presents a novel method for addressing moment closure problems by leveraging Gaussian processes to approximate Lagrange multipliers. The results demonstrate promising performance in various test cases, highlighting the potential of this data-driven approach for practical closure settings.