Neural operator preconditioning from mixed dataset for the Helmholtz equations: Application to transcranial ultrasound

Abstract

This work develops a neural operator preconditioned subspace method for sequences of linear systems arising from the discretization of the two-dimensional Helmholtz equation in transcranial ultrasound applications. The problem involves strongly heterogeneous, patient-dependent velocity fields that induce severe wave distortion and pose significant challenges for standard iterative solvers. Building on neural network preconditioning framework of Giraud et al. (HAL RR-9593, 2025) and the idealized skull dataset used for the learned optimizer of Stanziola et al. (JCP 441, 2021), neural operator preconditioners are trained on six mixed velocity-source datasets combining randomized source configurations and idealized skull-based velocity fields with random noise. The proposed mixed-dataset strategy aims to improve both computational efficiency and generalization across varying configurations. The neural operator is trained on a coarse grid using a physics-informed loss based on the relative residual of the discrete Helmholtz equation and is incorporated as a nonlinear preconditioner within flexible GMRES (FGMRES). Numerical experiments demonstrate that the resulting hybrid method efficiently solves practical transcranial ultrasound problems on grids 64 times larger than those used during training, whereas both classical GMRES and the learned optimizer fail to converge within comparable computational budgets. Moreover, the proposed method achieves arbitrary solution accuracies and exhibits strong out-of-distribution generalization across diverse source and velocity configurations. This work highlights the importance of dataset design in scientific machine learning and provides a practical framework for integrating matrix-free neural operator preconditioning with Krylov subspace methods for solving practical large-scale Helmholtz problems.