2017

Deep Residual Learning and PDEs on Manifold

Li, Zhen, Shi, Zuoqiang

Understand

In this paper, we formulate the deep residual network (ResNet) as a control problem of transport equation.

In ResNet, the transport equation is solved along the characteristics.
Based on this observation, deep neural network is closely related to the control problem of PDEs on manifold.
We propose several models based on transport equation and Hamilton-Jacobi equation.

Algorithms for overcoming the curse of dimensionality for certain hamilton-jacobi equations arising in control theory and elsewhere
J. Darbon and S. Osher · 2015
Earlier work this paper cites.
Splitting enables overcoming the curse of dimensionality
J. Darbon and S. Osher · 2015
Earlier work this paper cites.
Algorithm for overcoming the curse of dimensionality for certain non-convex hamilton-jacobi equations, projections and differential games
Y. T. Chow, J. Darbon, S. Osher, and W. Yin · 2016
Earlier work this paper cites.
Algorithm for overcoming the curse of dimensionality for time-dependent non-convex hamilton-jacobi equations arising from optimal control and differential games problems
Y. T. Chow, J. Darbon, S. Osher, and W. Yin · 2016
Earlier work this paper cites.

Deep residual learning for image recognition
K. He, X. Zhang, S. Ren, and J. Sun · 2016
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Identity mappings in deep residual networks
Original
K. He, X. Zhang, S. Ren, and J. Sun · 2016
Cited alongside, same era.
Algorithm for overcoming the curse of dimensionality for state-dependent hamilton-jacobi equations
Y. T. Chow, J. Darbon, S. Osher, and W. Yin · 2017
Cited alongside, same era.
Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning
Original
J. Han, A. Jentzen, and W. E · 2017
Cited alongside, same era.

Densely connected convolutional networks
G. Huang, Z. Liu, L. V. Maaten, and K. Q. Weinberger · 2017
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Point integral method for elliptic equations with variable coefficients on point cloud
Z. Li, Z. Shi, and J. Sun · 2017
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Weighted nonlocal laplacian on interpolation from sparse data
Z. Shi, S. Osher, and W. Zhu · 2017
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Deep neural networks with data dependent implicit activation function
B. Wang, X. Luo, Z. Li, W. Zhu, Z. Shi, and S. Osher · 2018
Closest in time.

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