Novel approach to solving Schwarzschild black hole perturbation equations via physics informed neural networks

Abstract

Machine learning, particularly neural networks, has rapidly permeated most activities and work where data has a story to tell. Recently, deep learning has started to be used for solving differential equations with input from physics, also known as Physics-Informed Neural Network (PINNs). Physics-Informed Neural Networks (PINNs) applications in numerical relativity remain mostly unexplored. To remedy this situation, we present the first study of applying PINNs to solve in the time domain the Zerilli and the Regge-Wheeler equations for Schwarzschild black hole perturbations. The fundamental difference of our work with other PINN studies in black hole perturbations is that, instead of working in the frequency domain, we solve the equations in the time domain, an approach commonly used in numerical relativity to study initial value problems. To evaluate the accuracy of PINNs results, we compare the extracted quasi-normal modes with those obtained with finite difference methods. For comparable grid setups, the PINN results are similar to those from finite difference methods and differ from those obtained in the frequency domain by a few percent. As with other applications of PINNs for solving partial differential equations, the efficiency of neural networks over other methods emerges when applied to large dimensionality or high complexity problems. Our results support the viability of PINNs in numerical relativity, but more work is needed to assess their performance in problems such as the collision of compact objects.