Green multigrid network

GreenLearning networks (GL) [Boulle et al., Sci. Rep. 12, 4824 (2022)] directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output neural network so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves 3.8%–39.15% accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about 10% of the full grid data, resulting in a 55.9% and 92.5% reduction in training time and Graphics processing unit (GPU) memory cost for one-dimensional test problems, and a 37.7% and 62.5% reduction for two-dimensional test problems.