Probing String Theory Compactifications via Geometry-Aware Neural Spectral Solvers

Quantum dynamics on the compact extra dimensions predicted by string theory plays a central role in determining the low-energy particle spectrum and coupling constants of effective four-dimensional theories. Yet direct numerical simulation of the time-dependent Schrodinger equation on six-dimensional Calabi–Yau manifolds remains computationally intractable with classical finite-element or finite-difference methods due to the exponential scaling of grid-based discretisations with spatial dimension. Here we introduce NeuralSpecSolve, a geometry-aware neural PDE framework that encodes Riemannian structure directly into the learning architecture via the spectral basis of the Laplace–Beltrami operator. Wavefunction amplitudes are represented as linear combinations of manifold eigenfunctions, and a physics-informed neural network—augmented with a Bayesian weight posterior for uncertainty quantification—learns the time evolution of spectral coefficients, bypassing explicit spatial discretisation. We demonstrate, on geometries ranging from S 2 and hyperbolic manifolds to synthetic Calabi–Yau-like spaces, that NeuralSpecSolve achieves L2 wavefunction errors below 3 × 10−4, recovers Kaluza–Klein mass spectra within 1% of analytical reference values, and reduces inference time by more than two orders of magnitude relative to classical finite-element solvers. A Bayesian extension quantifies predictive uncertainty arising from truncated spectral bases and sparse training data. Ablation studies and spectral mode analysis reveal energy-cascade behaviour intrinsic to curved-manifold quantum dynamics. Our results establish neural spectral methods as a scalable computational lens for exploring quantum physics in hidden dimensions and open new pathways for studying compactification landscapes in string theory.