Physics-Informed Neural Networks for the Relativistic Burgers Equation in the Exterior of a Schwarzschild Black Hole

TL;DR

This paper presents a physics-informed neural network designed to solve the relativistic Burgers equation around a Schwarzschild black hole, capable of modeling shock waves and discontinuities in curved spacetime.

Contribution

The paper introduces a novel PINN architecture with shock-aware features and Godunov-inspired residuals for simulating relativistic fluid dynamics near black holes.

Findings

Successfully models shock wave formation in curved spacetime

Reproduces smooth and discontinuous solutions accurately

Validates approach with various initial conditions

Abstract

We introduce a Physics-Informed Neural Networks(PINN) to solve a relativistic Burgers equation in the exterior domain of a Schwarzschild black hole. Our main contribution is a PINN architecture that is able to simulate shock wave formations in such curved spacetime, by training a shock-aware network block and introducing a Godunov-inspired residuals in the loss function. We validate our method with numerical experiments with different kinds of initial conditions. We show its ability to reproduce both smooth and discontinuous solutions in the context of general relativity.