Abstract
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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Barron A R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans Inform Theory, 1993, 39: 930–945
Bottou L, Curtis F E, Nocedal J. Optimization methods for large-scale machine learning. SIAM Rev, 2018, 60: 223–311
Chang B, Meng L, Haber E, et al. Multi-level residual networks from dynamical systems view. ArXiv:1710.10348, 2017
Chen Y, Li J, Xiao H, et al. Dual path networks. In: Advances in Neural Information Processing Systems, vol. 30. Long Beach: Neural Information Processing Systems Foundation, 2017, 4467–4475
Cybenko G. Approximation by superpositions of a sigmoidal function. Math Control Signals Systems, 1989, 2: 303–314
Deng J, Dong W, Socher R, et al. Imagenet: A large-scale hierarchical image database. In: Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition. Long Beach: IEEE, 2009, 248–255
E W. A proposal on machine learning via dynamical systems. Commun Math Stat, 2017, 5: 1–11
E W, Wang Q. Exponential convergence of the deep neural network approximation for analytic functions. Sci China Math, 2018, 61: 1733–1740
Ellacott S W. Aspects of the numerical analysis of neural networks. Acta Numer, 1994, 3: 145–202
Golub G H, Van Loan C F. Matrix Computations, 3rd ed. Baltimore: Johns Hopkins University Press, 2012
Gomez A N, Ren M, Urtasun R, et al. The reversible residual network: Backpropagation without storing activations. In: Advances in Neural Information Processing Systems, vol. 30. Long Beach: Neural Information Processing Systems Foundation, 2017, 2214–2224
Goodfellow I, Bengio Y, Courville A. Deep Learning. Cambridge: MIT Press, 2017
Haber E, Ruthotto L, Holtham E. Learning across scales—A multiscale method for convolution neural networks. ArXiv:1703.02009, 2017
Hackbusch W. Iterative Solution of Large Sparse Systems of Equations. New York: Springer, 1994
Hackbusch W. Multi-grid Methods and Applications. Heidelberg: Springer, 2013
He J, Li L, Xu J, et al. ReLU deep neural networks and linear finite elements. ArXiv:1807.03973, 2018
He K, Zhang X, Ren S, et al. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In: Proceedings of the 2015 IEEE International Conference on Computer Vision. Santiago: IEEE, 2015, 1026–1034
He K, Zhang X, Ren S, et al. Deep residual learning for image recognition. In: Proceedings of the 29th IEEE Conference on Computer Vision and Pattern Recognition. Las Vegas: IEEE, 2016, 770–778
He K, Zhang X, Ren S, et al. Identity mappings in deep residual networks. In: Proceedings of the 14th European Conference on Computer Vision. Amsterdam: Springer, 2016, 630–645
Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Networks, 1989, 2: 359–366
Hsieh J T, Zhao S, Eismann S, et al. Learning neural PDE solvers with convergence guarantees. In: Proceedings of the 7th International Conference on Learning Representations. https://openreview.net/forum?id=rklaWn0qK7, 2019
Huang G, Liu Z, Van Der Maaten L, et al. Densely connected convolutional networks. In: Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition. Honolulu: IEEE, 2017, 4700–4708
Ioffe S, Szegedy C. Batch normalization: Accelerating deep network training by reducing internal covariate shift. ArXiv:1502.03167, 2015
Katrutsa A, Daulbaev T, Oseledets I. Deep multigrid: Learning prolongation and restriction matrices. ArXiv:1711.03825, 2017
Ke T W, Maire M, Stella X Y. Multigrid neural architectures. ArXiv:1611.07661, 2016
Krizhevsky A, Hinton G. Learning multiple layers of features from tiny images. Technical report. Toronto: University of Toronto, 2009
Krizhevsky A, Sutskever I, Hinton G E. Imagenet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems, vol. 25. Lake Tahoe: Neural Information Processing Systems Foundation, 2012, 1097–1105
Larsson G, Maire M, Shakhnarovich S. Fractalnet: Ultra-deep neural networks without residuals. ArXiv:1605.07648, 2016
LeCun L, Bottou L, Bengio Y, et al. Gradient-based learning applied to document recognition. In: Proceedings of the IEEE, vol. 86. New York: IEEE, 1998, 2278–2324
LeCun Y, Bengio Y, Hinton G. Deep learning. Nature, 2015, 521: 436
Li Z, Shi Z. A flow model of neural networks. ArXiv:1708.06257v2, 2017
Lin T Y, Dollar P, Girshick R, et al. Feature pyramid networks for object detection. In: Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition. Honolulu: IEEE, 2017, 2117–2125
Liu D, Wen B, Liu X, et al. When image denoising meets high-level vision tasks: A deep learning approach. ArXiv:1706.04284, 2017
Long Z, Lu Y, Dong B. PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. ArXiv:1812.04426, 2018
Long Z, Lu Y, Ma X, et al. PDE-Net: Learning PDEs from data. In: Proceedings of the 35th International Conference on Machine Learning. Stockholm: PMLR, 2018, 3214–3222
Lu Y, Zhong A, Li Q, et al. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In: Proceedings of the 35th International Conference on Machine Learning. Stockholm: PMLR, 2018, 3282–3291
Mao X, Shen C, Yang Y B. Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections. In: Advances in Neural Information Processing Systems, vol. 29. Barcelona: Neural Information Processing Systems Foundation, 2016, 2802–2810
Milletari F, Navab N, Ahmadi S A. V-net: Fully convolutional neural networks for volumetric medical image segmentation. In: Proceedings of 2016 4th International Conference on 3D Vision. Stanford: IEEE, 2016, 565–571
Montanelli H, Du Q. Deep ReLU networks lessen the curse of dimensionality. ArXiv:1712.08688, 2017
Nair V, Hinton G E. Rectified linear units improve restricted boltzmann machines. In: Proceedings of the 27th International Conference on Machine Learning. Haifa: PMLR, 2010, 807–814
Noh H, Hong S, Han B. Learning deconvolution network for semantic segmentation. In: Proceedings of the 2015 IEEE International Conference on Computer Vision. Santiago: IEEE, 2015, 1520–1528
Pinkus A. Approximation theory of the MLP model in neural networks. Acta Numer, 1999, 8: 143–195
Ronneberger O, Fischer P, Brox T. U-net: Convolutional networks for biomedical image segmentation. In: Proceedings of Medical Image Computing and Computer-Assisted Intervention. Munich: Springer, 2015, 234–241
Shaham U, Cloninger A, Coifman R R. Provable approximation properties for deep neural networks. Appl Comput Harmon Anal, 2018, 44: 537–557
Siegel J W, Xu J. On the approximation properties of neural networks. ArXiv:1904.02311, 2019
Simonyan K, Zisserman A. Very deep convolutional networks for large-scale image recognition. ArXiv:1409.1556, 2014
Szegedy C, Liu W, Jia Y, et al. Going deeper with convolutions. In: Proceedings of the 28th IEEE Conference on Computer Vision and Pattern Recognition. Boston: IEEE, 2015, 1–9
Xu J. Iterative methods by space decomposition and subspace correction. SIAM Rev, 1992, 34: 581–613
Xu J. The Finite Element Methods. http://www.multigrid.org/wiki, 2019
Xu J, Zikatanov L. The method of alternating projections and the method of subspace corrections in Hilbert space. J Amer Math Soc, 2002, 15: 573–597
Xu J, Zikatanov L. Algebraic multigrid methods. Acta Numer, 2017, 26: 591–721
Zagoruyko S, Komodakis N. Wide residual networks. ArXiv:1605.07146, 2016
Zhang T, Qi G J, Xiao B, et al. Interleaved group convolutions. In: Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition. Honolulu: IEEE, 2017, 4373–4382
Zhang X, Li Z, Change Loy C, et al. Polynet: A pursuit of structural diversity in very deep networks. In: Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition. Honolulu: IEEE, 2017, 3900–3908
Zhou D X. Universality of deep convolutional neural networks. ArXiv:1805.10769, 2018
Acknowledgements
The first author was supported by the Elite Program of Computational and Applied Mathematics for PhD Candidates of Peking University. The second author was supported in part by the National Science Foundation of USA (Grant No. DMS-1819157) and the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program (Grant No. DE-SC0014400). The authors thank Xiaodong Jia for his help with the numerical experiments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, J., Xu, J. MgNet: A unified framework of multigrid and convolutional neural network. Sci. China Math. 62, 1331–1354 (2019). https://doi.org/10.1007/s11425-019-9547-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-019-9547-2