Deep Learning solutions to singular ordinary differential equations: from special functions to spherical accretion

TL;DR

This paper explores the use of Physics Informed Neural Networks (PINNs) to solve singular ordinary differential equations, demonstrating their effectiveness in handling singularities in equations relevant to physics.

Contribution

It introduces a novel application of PINNs to singular ODEs, including techniques like adaptive loss weighting to improve accuracy near singular points.

Findings

PINNs effectively approximate solutions across singularities.

Adaptive loss weighting enhances training accuracy.

Successful application to equations in gravitation and fluid dynamics.

Abstract

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.