标题： 通过多目标优化自动发现最优元求解器 显示英文标题

标题： Automatic discovery of optimal meta-solvers via multi-objective optimization

摘要： 我们设计了两类超快速元求解器，用于解决通过将神经算子与简单的迭代求解器（例如雅可比和高斯-赛德尔）或克雷洛夫子空间方法（例如GMRES和BiCGStab）结合离散偏微分方程后产生的线性系统，使用DeepONet的主干基作为粗粒度预条件器。 其思路是利用神经网络的频谱偏差来考虑误差分布中的低频部分，而高频部分则使用松弛方法或细粒度预条件器容易且廉价地处理。 我们使用多种指标创建了最优元求解器的帕累托前沿，并引入了一个偏好函数来选择最适合特定场景的最佳求解器。 这种寻找最优求解器的自动化方法可以扩展到非线性系统和其他设置，例如在时变偏微分方程中找到最佳的时空元求解器。 显示英文摘要

摘要： We design two classes of ultra-fast meta-solvers for linear systems arising after discretizing PDEs by combining neural operators with either simple iterative solvers, e.g., Jacobi and Gauss-Seidel, or with Krylov methods, e.g., GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse preconditioner. The idea is to leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled easily and inexpensively using relaxation methods or fine-scale preconditioners. We create a pareto front of optimal meta-solvers using a plurarilty of metrics, and we introduce a preference function to select the best solver most suitable for a specific scenario. This automation for finding optimal solvers can be extended to nonlinear systems and other setups, e.g. finding the best meta-solver for space-time in time-dependent PDEs.