Solving Einstein equations using deep learning

TL;DR

This paper introduces a neural network-based approach to numerically solve Einstein's field equations, successfully deriving key metrics like Schwarzschild and charged Schwarzschild, potentially advancing numerical relativity methods.

Contribution

It applies physics-informed neural networks to Einstein equations, providing a novel, flexible method for obtaining solutions in complex gravitational systems.

Findings

Successfully derived Schwarzschild metric

Obtained charged Schwarzschild metric

Demonstrated neural networks' potential in numerical relativity

Abstract

Einstein field equations are notoriously challenging to solve due to their complex mathematical form, with few analytical solutions available in the absence of highly symmetric systems or ideal matter distribution. However, accurate solutions are crucial, particularly in systems with strong gravitational field such as black holes or neutron stars. In this work, we use neural networks and auto differentiation to solve the Einstein field equations numerically inspired by the idea of physics-informed neural networks (PINNs). By utilizing these techniques, we successfully obtain the Schwarzschild metric and the charged Schwarzschild metric given the energy-momentum tensor of matter. This innovative method could open up a different way for solving space-time coupled Einstein field equations and become an integral part of numerical relativity.