Physics-Informed 3D Atomic Reconstruction and Dynamics of Free-Standing Graphene from Single Low-Dose TEM Images

Resolving the three-dimensional (3D) atomic geometry of free-standing graphene in real time is essential for understanding how intrinsic rippling governs its electronic properties. However, the low electron doses required to mitigate radiation damage impose severe signal-to-noise constraints that limit conventional reconstruction methods. Here, we present a physics-informed computational framework that reconstructs 3D atomic coordinates of single-layer graphene from individual low-dose transmission electron microscopy (TEM) frames (8x10^3 e-/Ang^2, 1 ms temporal resolution). The approach combines simulated annealing optimisation with molecular dynamics regularisation, achieving sub-angstrom out-of-plane accuracy (sigma_z < 0.45 Ang), validated against ground-truth simulations. A Kullback-Leibler divergence-based calibration aligns the forward model with experimental image statistics, reducing systematic bias. Applied to high-speed time-series data, the framework enables simultaneous extraction of real-time ripple dynamics, strain tensors, surface curvature, bond-length distributions, and density functional theory (DFT)-derived electron localisation functions (ELF). We establish quantitative relationships linking local geometry, strain, and bond-length variations to electron localisation, demonstrating that sub-angstrom structural fluctuations drive spatially localised, millisecond-scale electronic modulation. A critical dose threshold is identified below which structural information becomes irrecoverable, providing practical guidance for experimental design. The framework is broadly applicable to beam-sensitive two-dimensional materials.