Physics-Informed Neural Networks for Sparse Strain-Field Reconstruction in 4D-STEM

Quantitative strain mapping using four-dimensional scanning transmission electron microscopy (4D-STEM) typically requires densely sampled scans that can damage beam-sensitive specimens. We develop a physics-informed neural network (PINN) for sparse 4D-STEM strain reconstruction that embeds elastic equilibrium and Saint-Venant compatibility in the training loss through automatic differentiation. The architecture combines a coordinate-based implicit representation, sine activations with stable second derivatives, frozen residual-scale normalization, an exponential physics-weight ramp, and residual-based adaptive collocation. We apply a sine-activated residual network to an experimental $180\times400$-pixel strain map of domain-structured PbGeSnSe$_{1.5}$Te$_{1.5}$. Across $1$-$75$ sampling ($720$-$54{,}000$ probe positions), $R^2$ for $\varepsilon_{xx}$ reaches $0.80$ at $10$ sampling and saturates near $0.86$ by $25$; the chevron strain-band morphology is recovered from $10$ of probe positions. At $10$ sampling, the PINN reduces mean absolute error by approximately $26$ relative to compressed sensing and $22$ relative to Gaussian-process regression. An ablation against an equal-capacity data-only SIREN shows that the PDE prior improves accuracy at extreme sparsity and consistently improves physical self-consistency, but biases the reconstruction when data are abundant. Monte Carlo dropout and mean-field variational inference provide per-pixel epistemic uncertainty maps correlated with reconstruction error. With an appropriate constitutive model, the framework is adaptable to strain mapping across diverse material systems.