mlsanityData to 7 Oct 2026

Green Multigrid Network

arXiv 2407.03593 ↗First posted 4 Jul 2024math.NA cs.LG cs.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

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.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 36 of its 67 references that are in the map. First posted on arXiv 4 Jul 2024.

Builds on

36 of 67 references.

References · 36

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24 more
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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 2; 2026: 1.

3 papers.

  1. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  2. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  3. 2026IV-Net: A neural network for elliptic PDEs with random and highly varying coefficientscited by 0