AI and the World's Hard Problems · Physics · arXiv · 2026-10-21
Born-Preconditioned Learning for Iterative Neural PDE Solvers
Simulating high-frequency waves in spatially varying media remains challenging for both classical iterative methods and neural PDE solvers. Neural Preconditioned Born Series (NPBS) learns iterative neural PDE solvers by using a simple reference preconditioner to help learn a more effective neural one: building on the Convergent Born Series, a network receives a preconditioned residual and predicts a solution update, trained with a metric-matched objective that uses the same reference preconditioner in the input and the loss to reduce that residual after each update. On heterogeneous Helmholtz problems NPBS reduces iteration counts across neural architectures with larger gains at higher wavenumbers and achieves the lowest wall-clock time in the evaluated Krylov solver comparison, while extending naturally to three dimensions and improving convergence on convection-diffusion-reaction systems and Newton linear systems for nonlinear PDEs.