Neural Networks Assisted Metropolis-Hastings for Bayesian Estimation of Critical Exponent on Elliptic Black Hole Solution in 4D Using Quantum Perturbation Theory

TL;DR

This paper introduces a neural network-enhanced Bayesian Metropolis-Hastings method to accurately estimate the critical exponent in quantum perturbation theory of elliptic black hole solutions, accounting for measurement errors.

Contribution

It presents a novel neural network-assisted Bayesian algorithm for estimating critical exponents in quantum gravitational systems, improving error analysis over traditional methods.

Findings

Successfully identified the distribution of critical exponents.

Explored the range of physically distinguishable exponents.

Enhanced accuracy in quantum perturbation calculations.

Abstract

It is well-known that the critical gravitational collapse produces continuous self-similar solutions characterized by the Choptuik critical exponent, $\gamma$. We examine the solutions in the domains of the linear perturbation equations, considering the numerical measurement errors. Specifically, we study quantum perturbation theory for the four-dimensional Einstein-axion-dilaton system of the elliptic class of $\text{SL}(2,\mathbb{R})$ transformations. We develop a novel artificial neural network-assisted Metropolis-Hastings algorithm based on quantum perturbation theory to find the distribution of the critical exponent in a Bayesian framework. Unlike existing methods, this new probabilistic approach identifies the available deterministic solution and explores the range of physically distinguishable critical exponents that may arise due to numerical measurement errors.