Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning

In the boundary element method (BEM), the sinh transformation method is an effective method for evaluating nearly singular integrals, but a relationship between the integration accuracy and the number of Gaussian points is needed to achieve adaptive computation. Based on deep learning, we propose a novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), which can determine the number of Gaussian points according to the required accuracy. First, a large number of integration data samples of the sinh transformation method are generated in different cases, and the neural network is trained to establish the relationship between the number of Gaussian points and the integration accuracy. Then, based on the improved loss function and evaluation index, a better network model is obtained to ensure that the actual integration accuracy is slightly higher than the requirement of using the minimum Gaussian points. In this way, when the trained neural network is used in the sinh transformation method, the higher accuracy requirement can be met at a lower cost. Numerical examples demonstrate that, compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency when evaluating the nearly singular integrals for very thin coatings and other structures.