Addressing the gravitational collapse of a massless scalar field with Physics-Informed Neural Networks

Abstract

The gravitational collapse of a massless scalar field remains a demanding benchmark for numerical methods in numerical relativity, as it exhibits critical behavior at the boundary between dispersion and black hole formation. In this work we revisit this problem by relying on Physics-Informed Neural Networks (PINNs) as flexible solvers for partial differential equations, thereby providing a comparative assessment of several recent neural architectures. Building on the Einstein-massless-Klein-Gordon formulation in polar-areal coordinates, we consider four initial-value problems encompassing subcritical, critical, and supercritical regimes and use high-resolution finite-difference simulations as reference solutions. Our study is primarily comparative: we evaluate several state-of-the-art deep learning architectures, including vanilla and high-precision PINNs, sinusoidal-feature and quadratic-residual variants, and Kolmogorov-Arnold Networks, all trained under a common loss design that encodes the field equations, boundary conditions, and causal time-space enforcement, together with a novel adaptive spacetime sampling. Within this framework we also introduce ModPINN, a modest modification of standard PINNs that augments standard multilayer perceptrons with coordinate embeddings, quadratic layers, and other common ingredients in recent literature. This study shows that deep-learning-based methods can reproduce finite-difference solutions for the scalar field and the spacetime metric with competitive accuracy using significantly fewer collocation points than more traditional methodologies. While no single architecture dominates in all regimes, ModPINN achieves particularly stable and accurate solutions near criticality, indicating that suitably designed embeddings and adaptive sampling can enhance the robustness of PINNs for challenging gravitational-collapse scenarios.