Beyond the Final Iterate: Learning from Solver Trajectories for Inverse Medium Scattering

Inverse medium scattering concerns the reconstruction of spatially varying media from limited and noisy wave measurements. We investigate whether the trajectory generated by a classical solver can itself serve as a useful representation for learned inverse reconstruction. We propose a Transformer-based neural operator that learns reconstruction corrections from function-space trajectories generated by Gauss–Newton (GN) iterations with total variation (TV) regularization. An attention mechanism with convolutional query, key, and value projections aggregates information across the trajectory while preserving spatial structure. For Helmholtz inverse medium reconstruction with nominal $3\%$ observation noise, our method achieves a mean relative $L^2$ reconstruction error of $9.26\%$, compared with $13.96\%$ for GN–TV and $12.70\%$ for a comparably sized Fourier Neural Operator (FNO) baseline that corrects the final GN–TV iterate. Through extensive comparative experiments and ablation studies, we demonstrate the effectiveness of the proposed model. Without retraining, it achieves lower mean medium reconstruction errors than GN–TV across the tested noise levels and spatial resolutions, supporting its robustness to observation noise and discretization changes. These findings highlight solver trajectories as an effective interface between model-based inversion and operator learning.