arXiv:2603.18527v5 Announce Type: replace Abstract: Simulating high-frequency waves in spatially varying media remains challenging for both classical iterative methods and neural PDE solvers. We propose Neural Preconditioned Born Series (NPBS), a framework for learning iterative neural PDE solvers. Our guiding idea is to use a simple reference preconditioner to help learn a more effective neural preconditioner. Building on the Convergent Born Series (CBS), NPBS feeds a preconditioned residual to a neural network that predicts a solution update. We train the network to reduce the preconditioned residual after each update. The same reference preconditioner is used for the network input and the loss, giving a metric-matched training objective. Experiments on heterogeneous Helmholtz problems show reduced iteration counts across neural architectures and larger gains at higher wavenumbers. Within a Krylov solver, NPBS achieves the lowest wall-clock time in the evaluated solver comparison. The framework also extends naturally to three-dimensional problems. The same metric-matched formulation also improves convergence on convection-diffusion-reaction systems and Newton linear systems for nonlinear PDEs, indicating that NPBS provides a general framework for learning neural preconditioners.