Boundary Knot Neural Networks for the Inverse Cauchy Problem of the Helmholtz Equation

The traditional boundary knot method (BKM) has certain advantages in solving Helmholtz equations, but it still faces the difficulty of solving ill-posed problems when dealing with inverse problems. This work proposes a novel deep learning framework, the boundary knot neural networks (BKNNs), for solving inverse Cauchy problems of the Helmholtz equation. The method begins by uniformly distributing collocation points on the physical boundary, then employs a fully connected neural network to approximate the source point coefficient vector in the BKM. The physical quantities on the computational domain can be expressed by the BKM formula, and the loss functions can be constructed via accessible conditions on measurable boundaries. After that, the optimal weights and biases can be obtained by training the fully connected neural network, and thus, the source point coefficient vector can be successfully solved. As a machine learning-based meshless scheme, the BKNN eliminates tedious procedures like meshing and numerical integration while handling inverse Cauchy problems with complex boundaries. More importantly, the method itself is an optimization algorithm that completely avoids the complex processing techniques for ill-conditioned problems in traditional methods. Numerical experiments validate the efficacy of the proposed method, showcasing its superior performance over the traditional BKM for solving the Helmholtz equation’s inverse Cauchy problems.