The model and data for HXI-PINN

Based on this theory, a physics-constrained inversion network, HXI-PINN, is constructed by explicitly embedding the forward equation into the network architecture and optimization: the output layer enforces strict zero-error energy closure through ReLU non-negativity and counts-mean rescaling, while the loss function is dominated by counts-domain constraints (total weight 0.93) to approximate the full-channel forward response.

Three-level experiments demonstrate that HXI-PINN achieves NRMSE below 0.01 and SSIM above 0.98 on Gaussian sources, significantly and stably outperforming HXI-DLA, which degrades sharply with increasing morphological complexity; double-source dynamic-range tests (peak ratios 1:1--1:30) consistently achieve double-peak detection with counts-shape correlation above 0.9993 and zero energy-closure error, resolving two independent sources at the $\sim$3\arcsec\ instrumental resolution limit; on temporally independent soft X-ray complex morphologies, the network successfully reconstructs complex loop and extended flare source structures, verifying cross-morphology generalization; on the HXI observation event (2024 March 23), re-projected counts consistency slightly exceeds the CLEAN baseline, and the reconstructed bright region is overall consistent with SDO/AIA multi-band observations.

This paper systematically introduces physics-informed neural network (PINN) principles into HXI bi-grid modulation imaging, replacing pure data fitting with executable hard constraints that endow the reconstruction with mathematical-level physical consistency guarantees, and demonstrates that ``physical constraints + deep prior'' is an effective paradigm for underdetermined inversion---constraints anchor the solution in the feasible subspace satisfying the forward equation, while the deep prior selects the optimal solution within it.

The framework is in principle transferable to other modulation-imaging instruments such as Solar Orbiter/STIX and Yohkoh/HXT; its deterministic, millisecond-level inference and energy-conservation properties are suitable for real-time flare monitoring and batch processing.