| Abstract: |
| We propose a framework of the Green Multigrid network (GreenMGNet), a type of operator learning algorithm for a class of asymptotically smooth Green functions. The new framework presents itself better accuracy and efficient computational complexity, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, the Green function is modeled as a piecewise function to preserve its singular behavior in some part of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN), so that it can capture singularity accurately. Second, the asymptotic smoothness property of the Green function is used to leverage Multi-Level Multi-Integration(MLMI) algorithm for both training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectivity of the proposed method. On average, GreenMGNet achieves 3.8% to 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 GPU memory cost for one-dimensional test problems, and a 37.7% and 62.5% reduction for two-dimensional test problems. |
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