G-RANS: Generalizable Residual-Aware Neural Solvers for Sparse Systems

Neural operators have shown promise in accelerating PDE solvers, yet they remain unreliable for the sparse linear systems induced by discretization due to limited generalization across physical parameters and insufficient accuracy, and hybrid neural iterative schemes face stagnation as the residual distribution evolves over iterations. To address these limitations, we propose G-RANS (Generalizable Residual-Aware Neural Solver), a neuralized iterative paradigm that performs residual-aware subspace corrections by mapping the residual to the matrix graph, generating multi-scale correction subspaces via a residual-aware basis generator, and applying projected updates through a differentiable subspace projection. G-RANS is trained fully self-supervised via progressive bootstrap with multi-stage residual distributions. On sparse FEM systems from a representative suite of second-order elliptic PDEs (Poisson, advection--diffusion, reaction--diffusion, and Helmholtz), G-RANS is robust to severe coefficient shifts (up to $\pm70\\%$) and shows strong cross-equation generalization. G-RANS reaches relative residuals on the order of $10^{-5}$, substantially outperforming end-to-end neural operators, and attains the same target residual with $2$--$4\times$ less wall-clock time than classical Krylov solvers.