Abstract:

In popular machine learning approaches, a key problem involves learning a linear operator and its inverse. Given a complete orthogonal basis, this learning task can be reformulated as learning the expansion coefficients in the chosen basis. Within this framework, we propose a learning-based numerical method (LbNM) for solving the Helmholtz equation at high frequencies. The primary innovation lies in applying Tikhonov regularization to stably learn the solution operator by leveraging essential information, particularly from fundamental solutions. This operator can then be applied to new boundary inputs to swiftly update the solution. Using the method of fundamental solutions and quantitative Runge approximation, we establish an error estimate that demonstrates the interpretability and generalizability of our method. Numerical results validate this error analysis, showcasing the method’s high accuracy and efficiency.