We present a novel architecture for learning geometry-aware preconditioners for linear partial differential equations (PDEs). We show that a deep operator network (Deeponet) can be trained on a simple geometry and remain a robust preconditioner for problems defined by different geometries without further fine-tuning or additional data mining. We demonstrate our method for the Helmholtz equation, which is used to solve problems in electromagnetics and acoustics; the Helmholtz equation is not positive definite, and with absorbing boundary conditions, it is not symmetric.

翻译：我们提出了一种新颖的架构，用于学习线性偏微分方程的几何感知预处理器。我们证明，深度算子网络可以在简单几何结构上训练，并在无需进一步微调或额外数据挖掘的情况下，对不同几何结构定义的问题保持鲁棒的预处理性能。我们在亥姆霍兹方程上验证了该方法，该方程用于解决电磁学和声学问题；亥姆霍兹方程非正定，且在吸收边界条件下非对称。