Numerical G₂ Metric Construction via Physics-Informed Neural Networks: A Local Model of K₇ Neck Geometry

We develop a physics-informed neural network (PINN) framework to approximate torsion-free G₂-structures in seven dimensions, using a Cholesky parameterization of the metric with analytical warm-start. We first validate the approach on the flat 7-torus T⁷, where the method recovers the expected isotropic solution under a prescribed determinant constraint and yields residual torsion well within the small-torsion regime required by Joyce-style deformation results. A multi-seed study (5 independent initializations) shows convergence to the same solution to 9 significant digits. Spectral analysis via Monte Carlo Galerkin establishes baseline eigenvalue structure on T⁷ and demonstrates that local G₂ constraints alone cannot determine global spectral properties.We then introduce a 3-chart atlas prototype inspired by the Twisted Connected Sum (TCS) construction, enforcing overlap consistency — including a non-trivial Kovalev twist — via Schwarz-type iteration. The atlas achieves machine-precision interface matching and produces a qualitatively different low-lying Laplace spectrum compared to the single-chart torus baseline. The Cholesky warm-start technique — initializing the metric at an analytical target and learning only residual perturbations — may be of independent interest for other special-holonomy problems. We discuss in detail why local geometry alone cannot encode global topology, and why a multi-chart approach is necessary.