Deep Learning-Meshless Method for Inverse Potential Problems

This paper combines the deep learning method with the meshless method to propose a new numerical method, which is the deep learning-improved element-free Galerkin (DL-IEFG) method, for solving inverse potential problems. In this method, the unknown term in the governing equation of the inverse potential problem is represented by the feedforward neural network (FNN). By employing the improved element-free Galerkin (IEFG) method to solve the inverse potential problem, the solution equations are established to obtain numerical solutions. The training set is constructed on the valid values obtained from discretized observation spatial points. The predicted values at the training sample points are calculated by combining the numerical solutions with the approximation function built by the improved moving least-squares (MLS) approximation. Then, the FNN representing the unknown term is iterated using the loss function. The effectiveness of the DL-IEFG method for solving potential inverse problems is validated through numerical examples. In addition, the factors impacting the calculation accuracy and efficiency of the DL-IEFG method are investigated.