PhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion

Coefficient inversion on fine grids under PDE constraints is ill conditioned: sparse observations weakly constrain fine-scale parameters, and direct single-resolution optimization must recover state and coefficient fields across all scales simultaneously. This causes slow, initialization-sensitive convergence; learned transfer models require offline data and can introduce approximation error into the numerical physics. We propose PhyRes-MDNF, a fixed-physics multilevel discrete neural field framework. On each level, a single-level DNF represents the inverse unknowns directly as trainable fields and optimizes the discrete objective. In the full-space Darcy realization, state fields $U$ and their shared coefficient field $K$ are optimized jointly in one fixed-physics inverse process. Between levels, one zero-initialized PhyRes-GNN jointly performs fixed-stencil prolongation and bounded residual correction to construct an incoming target representation, which a fixed initialization map converts to the next DNF variables. It is fitted anew from the observations and unchanged numerical model, without offline pretraining or fine-grid truth. Coarse levels therefore resolve large-scale structure before refined degrees of freedom are introduced, shortening the fine-grid optimization path while retaining the original discrete operator. Under the same final-grid update budget, the multilevel Darcy realization reduces coefficient and state errors by approximately $85\%$ and $90\%$, respectively, demonstrating improved accuracy and final-grid iteration efficiency. On measured KTC2023 EIT data, the full-$W$ pipeline improves mean Otsu mIoU by approximately $3.4\%$ over the official linearized CEM reconstruction and $16.9\%$ over direct single-level DNF.