Sequential Monte Carlo with Cross-validated Neural Networks for Complexity of Hyperbolic Black Hole Solutions in 4D

TL;DR

This paper introduces a novel Bayesian Sequential Monte Carlo method combined with neural networks to accurately estimate complex, multi-modal solutions of hyperbolic black hole equations in 4D, accounting for measurement errors.

Contribution

It develops a new probabilistic framework using SMC and neural networks for solving highly nonlinear, multi-modal equations in gravitational physics, incorporating measurement errors and cross-validation.

Findings

Confirmed known solutions in the literature.

Identified all possible solutions considering measurement errors.

Enhanced prediction of critical functions for multiple solutions.

Abstract

This paper investigates the self-similar solutions of the Einstein-axion-dilaton configuration from type IIB string theory and the global SL(2,R) symmetry. We consider the Continuous Self Similarity (CSS), where the scale transformation is controlled by an SL(2, R) boost or hyperbolic translation. The solutions stay invariant under the combination of space-time dilation with internal SL(2,R) transformations. We develop a new formalism based on Sequential Monte Carlo (SMC) and artificial neural networks (NNs) to estimate the self-similar solutions to the equations of motion in the hyperbolic class in four dimensions. Due to the complex and highly nonlinear patterns, researchers typically have to use various constraints and numerical approximation methods to estimate the equations of motion; thus, they have to overlook the measurement errors in parameter estimation. Through a Bayesian framework, we incorporate measurement errors into our models to find the solutions to the hyperbolic equations of motion. It is well known that the hyperbolic class suffers from multiple solutions where the critical collapse functions have overlap domains for these solutions. To deal with this complexity, for the first time in literature on the axion-dilaton system, we propose the SMC approach to obtain the multi-modal posterior distributions. Through a probabilistic perspective, we confirm the deterministic $\alpha$ and $\beta$ solutions available in the literature and determine all possible solutions that may occur due to measurement errors. We finally proposed the penalized Leave-One-Out Cross-validation (LOOCV) to combine the Bayesian NN-based estimates optimally. The approach enables us to determine the optimum weights while dealing with the co-linearity issue in the NN-based estimates and better predict the critical functions corresponding to multiple solutions of the equations of motion.