Physics-informed neural network for material identification via distortion-robust polychromatic x-ray attenuation correction in photon-counting detectors

Spectral distortions in photon-counting detectors (PCDs) fundamentally limit the quantitative accuracy of material identification. While machine learning is used for compensation, current data-driven methods often lack physical constraints, limiting their interpretability and reliability across varying conditions. To address this issue, we propose a physics-informed neural network (PINN) framework that explicitly embeds the Beer–Lambert law into the learning architecture. By integrating an explicit differential layer to extract high-order curvature features from distorted spectra, the model enables direct inference of the effective atomic number and areal density. This approach effectively leverages the Z-dependent non-linear profile of the photoelectric effect, even when explicit absorption edges are outside the primary detection window. Simulation results establish a high-precision benchmark for Z_eff estimation in the target low-Z range (613), with an RMSE of 0.2111. Experimental validation on a CdZnTe-PCD further demonstrates that this accuracy improvement is preserved under realistic pulse pile-up and noise conditions, achieving an RMSE of 0.2457 and an R² of 0.9670. Compared with conventional physical correction methods (typically ±0.5 error margin), the proposed framework provides improved precision, with 92.86% of Z_eff estimation errors falling within ±0.4, corresponding to an approximately 20% tighter error bound. These results confirm that the proposed framework effectively mitigates spectral distortion, providing a robust, calibration-free solution for precise material identification of low-Z materials in industrial non-destructive testing.