Machine learning automorphic forms for black holes

Modular, Jacobi, and mock-modular forms serve as generating functions for BPS black hole degeneracies. By training feed-forward neural networks on Fourier coefficients of automorphic forms derived from the Dedekind eta function, Eisenstein series, and Jacobi theta functions, we demonstrate that machine learning techniques can accurately predict modular weights from truncated expansions. Our results reveal strong performance for negative weight modular and quasi-modular forms, particularly those arising in exact black hole counting formulae, with lower accuracy for positive weights and more complicated combinations of Jacobi theta functions. This study establishes a proof of concept for using machine learning to identify how data is organized in terms of modular symmetries in gravitational systems and suggests a pathway toward automated detection and verification of symmetries in quantum gravity.

Classical integrability in the presence of a cosmological constant: analytic and machine learning results

We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the presence of a potential for the neutral scalar fields. By focusing on a certain solution subspace, we show that a subset of the equations of motion in two dimensions are the compatibility conditions for a modified version of the Breitenlohner-Maison linear system. Subsequently, we study the Liouville integrability of the 2D models encoding the chosen 4D solution subspace from a one-dimensional point of view by constructing Lax pair matrices. In this endeavour, we successfully employ a linear neural network to search for Lax pair matrices for these models, thereby illustrating how machine learning approaches can be effectively implemented to augment the identification of integrable structures in classical systems.