MgNet swMATH ID: 35862 Software Authors: He, Juncai; Xu, Jinchao Description: MgNet: A unified framework of multigrid and convolutional neural network. We develop a unified model, known as MgNet, that simultaneously recovers some convolutional neural networks (CNN) for image classification and multigrid (MG) methods for solving discretized partial differential equations (PDEs). This model is based on close connections that we have observed and uncovered between the CNN and MG methodologies. For example, pooling operation and feature extraction in CNN correspond directly to restriction operation and iterative smoothers in MG, respectively. As the solution space is often the dual of the data space in PDEs, the analogous concept of feature space and data space (which are dual to each other) is introduced in CNN. With such connections and new concept in the unified model, the function of various convolution operations and pooling used in CNN can be better understood. As a result, modified CNN models (with fewer weights and hyperparameters) are developed that exhibit competitive and sometimes better performance in comparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets. Homepage: https://link.springer.com/article/10.1007/s11425-019-9547-2 Keywords: convolutional neural network; multigrid; unified framework; network architecture Related Software: Adam; PDE-Net; DeepONet; U-Net; DGM; D3M; ImageNet; GitHub; TensorFlow; torchdiffeq; PyTorch; AlexNet; PINNsNTK; DeepXDE; DiffSharp; V-Net; MeshCNN; Keras; Gmsh; XPINNs Cited in: 17 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year MgNet: a unified framework of multigrid and convolutional neural network. Zbl 1476.65026He, Juncai; Xu, Jinchao 2019 all top 5 Cited by 36 Authors 4 Dong, Suchuan 2 Dong, Bin 2 Li, Zongwei 2 Xu, Jinchao 2 Yang, Jielin 1 Arridge, Simon R. 1 Bordas, Stéphane Pierre Alain 1 Chen, Yuyan 1 Cheng, Xiuyuan 1 Cox, Ben T. 1 Deng, Wei 1 Deshpande, Saurabh A. 1 Fan, Yuwei 1 Gao, Han 1 He, Juncai 1 Lengiewicz, Jakub 1 Li, Yingzhou 1 Li, Zhihan 1 Lin, Guang 1 Lu, Jianfeng 1 Schwab, Christoph 1 Sheng, Hailong 1 Stanziola, Antonio 1 Stein, Andreas 1 Sun, Luning 1 Treeby, Bradley E. 1 Wang, Jianxun 1 Wang, Yating 1 Wang, Zhongjian 1 Wu, JinLong 1 Xiao, Heng 1 Xin, Jack X. 1 Ying, Lexing 1 Zeng, Yang 1 Zhang, Haimiao 1 Zhang, Zhiwen all top 5 Cited in 7 Serials 8 Journal of Computational Physics 3 Computer Methods in Applied Mechanics and Engineering 2 Communications in Computational Physics 1 Multiscale Modeling & Simulation 1 Science China. Mathematics 1 Journal of the Operations Research Society of China 1 Research in the Mathematical Sciences all top 5 Cited in 9 Fields 12 Numerical analysis (65-XX) 12 Computer science (68-XX) 4 Statistics (62-XX) 4 Fluid mechanics (76-XX) 3 Partial differential equations (35-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Biology and other natural sciences (92-XX) 1 Operator theory (47-XX) 1 Operations research, mathematical programming (90-XX) Citations by Year