Adaptive Neural Subspaces with Geometry-Aware Quadrature for Interface Problems

Abstract Based on the combination of neural networks, numerical quadrature schemes and adaptive subspace approximation, we propose a new machine learning method for solving interface problems. The neural network is employed to construct the basis functions of a finite-dimensional subspace, upon which the discrete solution is obtained via subspace approximation. In particular, an a posteriori error estimator can be derived from the hypercircle technique within this subspace approximation framework. This estimator can further be used as the loss function to adaptively refine the neural network parameters. The high accuracy achieved by the proposed method also demonstrates that machine learning approaches equipped with high precision integration are capable of automatically handling singularities arising in interface problems. Numerical results illustrate that the proposed method attains high accuracy for interface problem simulations when equipped with a high precision numerical integration scheme for designing loss functions.