Deep Learning Fractal Superconductivity: A Comparative Study of Physics-Informed and Graph Neural Networks Applied to the Fractal TDGL Equation

The fractal extension of the time-dependent Ginzburg–Landau (TDGL) equation, formulated within the framework of Scale Relativity, generalizes superconducting dynamics to non-differentiable space–time. Although analytically well established, its numerical solution remains difficult because of the strong coupling between amplitude and phase curvature. Here we develop two complementary deep learning solvers for the fractal TDGL (FTDGL) system. The Fractal Physics-Informed Neural Network (F-PINN) embeds the Scale-Relativity covariant derivative through automatic differentiation on continuous fields, whereas the Fractal Graph Neural Network (F-GNN) represents the same dynamics on a sparse spatial graph and learns local gauge-covariant interactions via message passing. Both models are trained against finite-difference reference data, and a parametric study over the dimensionless fractality parameter D quantifies its influence on the coherence length, penetration depth, and peak magnetic field. Across multivortex benchmarks, the F-GNN reduces the relative L2 error on ψ2 from 0.190 to 0.046 and on Bz from approximately 0.62 to 0.36 (averaged over three seeds). This ≈4× improvement in condensate-density accuracy corresponds to a substantial enhancement in vortex-core localization—from tens of pixels of uncertainty to sub-pixel precision—and yields a cleaner reconstruction of the 2π phase winding around each vortex, improving the extraction of experimentally relevant observables such as ξeff, λeff, and local Bz peaks. The model also preserves flux quantization and remains robust under 2–5% Gaussian noise, demonstrating stable learning under experimentally realistic perturbations. The D—scan reveals broader vortex cores, a non-monotonic variation in the penetration depth, and moderate modulation of the peak magnetic field, while preserving topological structure. These results show that graph-based learning provides a superior inductive bias for modeling non-differentiable, gauge-coupled systems. The proposed F-PINN and F-GNN architectures therefore offer accurate, data-efficient solvers for fractal superconductivity and open pathways toward data-driven inference of fractal parameters from magneto-optical or Hall-probe imaging experiments.