Learning Iteration Schedules, Not Inverses: Zero-Shot Amortized Neural Preconditioning for Large-Scale Linear Systems

Existing learned preconditioners are typically trained *per matrix*—with GNP requiring $2000$ steps ($61.6$s) per instance and GMP requiring $2000$ epochs ($133.9$s)—causing computational overhead to scale linearly with every solved system. Amortization fails because traditional approaches attempt to approximate the matrix inverse directly: a dense, domain-specific object of order $\mathcal{O}(n^2)$. We shift the target of learning. Krylov solvers never consume the inverse; instead, they operate on a *configuration* comprising a bounded row scaler paired with step sizes for a degree-$T$ residual polynomial. This target is low-dimensional, scale-free, and universally defined across matrix families, enabling zero-shot prediction rather than per-instance optimization. The resulted proposed method, Mamba-PREC, distills a single network on a corpus disjoint from the benchmark. At inference, it sweeps matrix rows in Reverse Cuthill–McKee order via a bidirectional selective state-space model, predicts a configuration, validates it through a blind probe on a synthetic right-hand side, and freezes it for the solve. On the $867$-matrix SuiteSparse benchmark, Mamba-PREC eliminates per-matrix training ($0.00$s vs. $61.6$s and $133.9$s) with a median setup time of $0.80$s, outperforms both baselines in time-to-tolerance and time-weighted AUC, and incurs zero non-finite breakdowns ($0$ vs. $13$ and $23$). On the $95$ largest systems in the collection—every structurally nonsymmetric real SuiteSparse matrix of order above $10^5$, the largest being $1.1\times 10^7$—where GNP and GMP fail to construct on $27$ and $43$ instances respectively, Mamba-PREC dominates all four headline metrics at a median solve time of $0.44$s. In pairwise comparisons on converged instances, Mamba-PREC reaches target tolerances faster in $98.9$% of runs against GMP, $82.4$% against GNP, $86.6$% against classical AMG, $65.9$% against Chebyshev semi-iteration, and $60.6$% against thresholded GPU ILU, yielding median time-to-tolerance speedups of $44.6\times$ (GMP), $23.4\times$ (GNP), $87.4\times$ (AMG), $2.8\times$ (Chebyshev), and $3.1\times$ (ILU).