GreenLearning networks (GL) 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(AugNN) 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 (MLMI) 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 $3.8\%$ 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.

翻译：绿色学习网络（GL）直接在物理空间中学习格林函数，使其成为捕捉偏微分方程未知解算子的可解释模型。对于许多偏微分方程，相应的格林函数表现出渐近光滑性。本文提出一个名为绿色多重网格网络（GreenMGNet）的框架，这是一种针对一类渐近光滑格林函数设计的算子学习算法。与开创性的GL相比，新框架在精度和效率上均有提升，实现了显著改进。GreenMGNet包含两项技术革新：首先，格林函数被建模为分段函数，以考虑其在超平面部分区域的奇异行为。该分段函数随后通过具有增强输出的神经网络（AugNN）进行逼近，从而准确捕捉奇异性。其次，利用格林函数的渐近光滑性，在训练和推理阶段均采用多级多积分（MLMI）算法。本文通过多个算子学习的测试案例验证了所提方法的精度和有效性。平均而言，GreenMGNet实现了3.8%至39.15%的精度提升。为达到GL的精度水平，GreenMGNet仅需约10%的全网格数据，在一维测试问题中训练时间和GPU内存成本分别降低55.9%和92.5%，在二维测试问题中分别降低37.7%和62.5%。