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

TL;DR

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.

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.

Figures (6)

• Figure 1: Diagrammatic representation of the adaptive model. The figure shows the landscape of a sample loss function with respect to the $(t,r)$ sampling, whose values span several orders of magnitude. The arrows indicate the direction of movement of the collocation points, which intuitively shift towards regions having higher PDE residuals.
• Figure 2: The proposed methodology for the ModPINN. The embedding block of the model modulates $(t,r)$ via a hyperbolic-tangent "warp", augments them with low-order polynomials of the normalized coordinates together with a dictionary of $\mathrm{M}$ Gaussian transforms, and projects the concatenated features through stacked quadratic (QRes) layers to output the physical set $(\phi,\alpha,C)$.
• Figure 3: Convergence of the relative $l_{2}$ errors for each physical variable (row by row) over training, across different models and initial-value problems. Beyond approximately $40,000$ epochs, performance gains become barely minimal.
• Figure 4: Relative total $l_{2}$ error of the proposed ModPINN for the considered scalar-field initial-value problems (by column) as a function of the hyperparameter $\epsilon_{\mathrm{t}}$, which modulates the strength of the temporal causality. Each row corresponds to a different spatiotemporal grid resolution $(t,r)$, and each sub-figure reports results for three network sizes.
• Figure 5: Top: $l_{2}$ errors for each variable, together with their variability across ten independent runs, shown for different models considered. Bottom: total $l_{2}$ errors as a function of the adaptive-sampling strength $\Lambda$, evaluated for each initial-condition problem. Only the ModPINN model is considered.