High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.

翻译：高频问题在偏微分方程的数值方法中一直是显著挑战。本文针对高频亥姆霍兹方程提出了一种基于学习的数值方法（LbNM）。其主要创新在于利用Tikhonov正则化方法，通过整合相关信息（特别是基本解）来稳定地学习解算子，进而将该解算子应用于新的边界输入，即可快速更新解。基于基本解法和定量Runge逼近，我们给出了误差估计，这表明了当前方法的可解释性和泛化能力。数值结果验证了误差分析，并展示了该方法的高精度和高效率特性。