Green Multigrid Network
Ye Lin, Young Ju Lee, Jiwei Jia

TL;DR
GreenMGNet is a novel operator learning framework that models Green's functions as piecewise functions with neural networks, leveraging asymptotic smoothness for improved accuracy and efficiency in solving PDEs.
Contribution
It introduces a piecewise neural network approach combined with MLMI to effectively learn asymptotically smooth Green's functions, outperforming previous GreenLearning methods.
Findings
Achieves 3.8% to 39.15% accuracy improvement.
Reduces training time by up to 55.9%.
Requires only 10% of full grid data for comparable accuracy.
Abstract
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…
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Taxonomy
TopicsDistributed and Parallel Computing Systems · Interconnection Networks and Systems
