Beyond Prediction: Extracting Preconditioning Structure from Fourier Neural Operators

Neural operators provide rapid predictions, while classical iterative solvers deliver high-accuracy solutions, motivating growing interest in combining their strengths. Interpreting neural operators as learned approximations to inverse PDE operators naturally suggests Jacobian-based preconditioning. However, solution supervision alone does not ensure effective preconditioners, leading existing approaches to modify training objectives or impose additional constraints. Our theoretical analysis shows that Fourier neural operators (FNOs) trained for solution prediction can still contain recoverable preconditioning structure. Building on this insight, we introduce purification, a post-processing technique that extracts this structure from FNO Jacobians without retraining. Experiments on Poisson problems support our theoretical analysis. For Helmholtz problems, NOP reduces mean GMRES iterations by up to $8.2\times$ compared with unpreconditioned GMRES. For nonlinear convection–diffusion problems, NOP reduces mean cumulative GMRES iterations by up to $9.6\times$ compared with unpreconditioned Newton–GMRES. These results offer a new perspective on how neural operators can enhance iterative solvers through structure extraction.