Physics-embedded neural computational electron microscopy for quantitative 4D nanometrology

Abstract: The fusion of rigorous physical laws with flexible data-driven learning represents a new frontier in scientific simulation, yet bridging the gap between physical interpretability and computational efficiency remains a grand challenge. In electron microscopy, this divide limits the ability to quantify three-dimensional topography from two-dimensional projections, fundamentally constraining our understanding of nanoscale structure-function relationships. Here, we present a physics-embedded neural computational microscopy framework that achieves metrological three-dimensional reconstruction by deeply coupling a differentiable electron-optical forward model with deep learning. By introducing a Vision Field Transformer as a high-speed, differentiable surrogate for physical process analysis simulations, we establish an end-to-end, self-supervised optimization loop that enforces strict physical consistency with hardware geometry. This synergy enables single-shot, quantitative three-dimensional nanometrology with precision comparable to atomic force microscopy but at orders of magnitude higher throughput. Furthermore, we demonstrate the capability for four-dimensional (3D real space plus time) in situ characterization by tracking the dynamic evolution of surface nanostructure during copper redox, revealing hidden crystallographic kinetics invisible to conventional imaging. Our work not only redefines the limits of scanning electron microscopy but also establishes a generalizable archetype for solving ill-posed inverse problems across physical sciences, unlocking the full potential of simulation as a third pillar of discovery.