AI 中文总结
针对经颅超声中二维亥姆霍兹方程离散化的线性系统问题，基于神经网络预处理框架和理想化头骨数据集，在混合数据集上训练神经算子预处理器，结合FGMRES方法，有效解决大规模问题，提高计算效率与泛化能力，突出数据集设计重要性。
AI 中文摘要
本文针对经颅超声应用中二维亥姆霍兹方程离散化产生的线性系统序列，开发了一种神经算子预处理子空间方法。该问题涉及强非均匀、依赖患者的速度场，给标准迭代求解器带来挑战。基于吉罗等人的神经网络预处理框架和斯坦齐奥拉等人的理想化头骨数据集，在六个混合速度 - 源数据集上训练神经算子预处理器。通过结合随机源配置、理想化头骨速度场和随机噪声，旨在提高计算效率和泛化能力。神经算子在粗网格上基于离散亥姆霍兹方程的相对残差使用物理信息损失进行训练，并纳入灵活广义最小残差法（FGMRES）作为非线性预处理器。数值实验表明，该混合方法能有效解决比训练时网格大64倍的实际经颅超声问题，而经典GMRES和学习优化器在可比计算预算内无法收敛。此外，该方法能达到任意解精度，并在不同源和速度配置下具有强分布外泛化能力。这项工作突出了数据集设计在科学机器学习中的重要性，并提供了一个实用框架，将无矩阵神经算子预处理与 Krylov子空间方法集成以解决实际大规模亥姆霍兹问题。
英文摘要
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.
