Learning Spectral Preconditioner for Linear System Solver with Application to Four-Dimensional Variational Data Assimilation

Four-dimensional variational data assimilation (4DVar) requires solving large-scale, ill-conditioned linear systems within its inner loop, where convergence depends on the spectral properties of the system matrix. Traditional limited-memory preconditioners (LMPs) reuse historical Lanczos/Ritz vectors but may lose effectiveness as the relevant subspace rotates between assimilation windows. We propose a neural spectral preconditioner that predicts a current-window subspace from historical features using a Fourier neural operator. The network output is orthonormalized before the preconditioner is constructed. Under the control-variable-transformed model, this constructed preconditioner is symmetric positive definite. Experiments on the strongly nonlinear Lorenz-96 system demonstrate that the proposed method reduces the number of conjugate gradient iterations by 54.16% compared with unpreconditioned CG and by 39.6% compared with LMP. On the quasi-geostrophic (QG) system, the method reduces the iteration count by 22.5% relative to LMP and improves fixed-iteration solution accuracy in the reported tests. The method retains the PCG solution framework and yields consistent reductions in iteration count across the evaluated dynamical regimes.