RINS: Residual-Image Neural Subspace Solvers for Large Sparse Linear Systems

Neural subspace solvers offer a promising way to accelerate the large sparse linear systems arising from PDE discretizations, but their value depends on generating effective correction directions at low computational overheads. We study this problem from a residual-image viewpoint: a correction subspace is useful when its image under the system operator captures the current residual. Guided by this view, we propose Gate-RINS, which generates multiple correction directions from shared polynomial residual probes and modulates them with a lightweight residual- and coordinate-dependent pointwise gate. We also introduce a hybrid schedule that uses GRANS-style graph controllers for the first correction block and Gate-RINS thereafter, combining an expressive graph based initial step with a lower-cost corrector as the solve progress. Theoretical analysis establishes a unified residual-image and dynamic approximate-inverse perspective for Gate-RINS and G-RANS, showing that a single neural-generated direction can emulate the residual quality of fixed-step GMRES. Numerical experiments on six PDE benchmarks demonstrate that Gate-RINS reaches a relative residual of $10^{-5}$ with 1.6-2.3$\times$ and 2.2-3.0$\times$ less wall-clock time than G-RANS and GMRES respectively. For Helmholtz systems with meshsize up to $10^5$, Gate-RINS also achieves the lowest residual and outperforms a GPU-accelerated parallel GMRES implementation, while GRANS fails to reach relative residual of $10^{-4}$ in the same time budget.