Fetching the paper…
Reading the bibliography…
While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge.
A multi-scale dnn algorithm for nonlinear elliptic equations with multiple scales,
X.-A. Li, Z.-Q. J. Xu, L. Zhang, · 1906
Earlier work this paper cites.
Anomalous slow diffusion from perpetual homogenization,
H. Owhadi, · 1969
Earlier work this paper cites.
G. Papanicolau, A. Bensoussan, J.-L. Lions, Asymptotic analysis for periodic structures, Elsevier, 1978
1978
Earlier work this paper cites.
J. L. Lions, A. Bensoussan, G. Papanicolaou, Asymptotic analysis for periodic structures, volume 5, North Holland, Amsterdam, 1978
1978
Earlier work this paper cites.
Iterative methods by space decomposition and subspace correction,
J. Xu, · 1992
Earlier work this paper cites.
A multiscale finite element method for elliptic problems in composite materials and porous media,
T. Y. Hou, X.-H. Wu, · 1997
Earlier work this paper cites.
The variational multiscale method—a paradigm for computational mechanics,
T. J. R. Hughes, G. Feijoo, L. Mazzei, J. Quincy, · 1998
Earlier work this paper cites.
The variational multiscale method—a paradigm for computational mechanics,
T. J. R. Hughes, G. R. Feijóo, L. Mazzei, J.-B. Quincy, · 1998
Earlier work this paper cites.
C. P. Robert, G. Casella, Monte Carlo Statistical Methods, 1999
1999
Earlier work this paper cites.
Multi-scale deep neural network (mscalednn) for solving poisson-boltzmann equation in complex domains,
Z. Liu, W. Cai, Z.-Q. J. Xu, · 2001
Earlier work this paper cites.
Heterogeneous multiscale methods,
W. E, B. Engquist, · 2003
Earlier work this paper cites.
Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models,
G. J. M. Hagelaar, L. C. Pitchford, · 2005
Earlier work this paper cites.
Analysis of the heterogeneous multiscale method for elliptic homogenization problems,
P. Ming, P. Zhang, et al., · 2005
Earlier work this paper cites.
Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows,
Y. Bazilevs, V. Calo, J. Cottrell, T. Hughes, A. Reali, G. Scovazzi, · 2007
Earlier work this paper cites.
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Numerical Mathematics, 2007
2007
Earlier work this paper cites.
Homogenization of Parabolic Equations with a Continuum of Space and Time Scales,
H. Owhadi, L. Zhang, · 2007
Earlier work this paper cites.
Homogenization of parabolic equations with a continuum of space and time scales,
H. Owhadi, L. Zhang, · 2008
Earlier work this paper cites.
Numerical homogenization of the acoustic wave equations with a continuum of scales,
H. Owhadi, L. Zhang, · 2008
Earlier work this paper cites.
Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast,
L. Berlyand, H. Owhadi, · 2010
Earlier work this paper cites.
Understanding the difficulty of training deep feedforward neural networks,
X. Glorot, Y. Bengio, · 2010
Earlier work this paper cites.
Free Energy Calculations by the Molecular Mechanics Poisson-Boltzmann Surface Area Method.,
N. Homeyer, H. Gohlke, · 2012
Earlier work this paper cites.
Adaptivity and variational stabilization for convection-diffusion equations,
A. Cohen, W. Dahmen, G. Welper, · 2012
Earlier work this paper cites.
V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer Science & Business Media, 2012
2012
Earlier work this paper cites.
L. Berlyand, A. G. Kolpakov, A. Novikov, Introduction to the network approximation method for materials modeling, volume 148, Cambridge University Press, 2013
2013
Earlier work this paper cites.
Generalized multiscale finite element methods,
Y. Efendiev, J. Galvis, T. Hou, · 2013
Earlier work this paper cites.
Polyharmonic homogenization, rough polyharmonic splines and sparse super-locatization,
H. Owhadi, L. Zhang, L. Berlyand, · 2014
Cited alongside, same era.
Localization of elliptic multiscale problems,
A. Målqvist, D. Peterseim, · 2014
Cited alongside, same era.
A localized orthogonal decomposition method for semi-linear elliptic problems,
P. Henning, A. Målqvist, D. Peterseim, · 2014
Cited alongside, same era.
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization,
H. Owhadi, L. Zhang, L. Berlyand, · 2014
Cited alongside, same era.
Deep learning,
Y. LeCun, Y. Bengio, G. Hinton, · 2015
Cited alongside, same era.
Bayesian numerical homogenization,
H. Owhadi, · 2015
Cited alongside, same era.
Theory of the Frequency Principle for General Deep Neural Networks,
T. Luo, Z. Ma, Z.-Q. J. Xu, Y. Zhang, · 2019
Later among the works it cites.
The convergence rate of neural networks for learned functions of different frequencies,
B. Ronen, D. Jacobs, Y. Kasten, S. Kritchman, · 2019
Later among the works it cites.
Machine learning from a continuous viewpoint,
W. E, C. Ma, L. Wu, · 2019
Later among the works it cites.
Towards understanding the spectral bias of deep learning,
Y. Cao, Z. Fang, Y. Wu, D.-X. Zhou, Q. Gu, · 2019
Later among the works it cites.
A fine-grained spectral perspective on neural networks,
G. Yang, H. Salman, · 2019
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
I. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT press, Cambridge, 2016
2016
Cited alongside, same era.
Deep residual learning for image recognition,
K. He, X. Zhang, S. Ren, J. Sun, · 2016
Cited alongside, same era.
A Proposal on Machine Learning via Dynamical Systems,
W. E, · 2017
Cited alongside, same era.
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations,
W. E, J. Han, A. Jentzen, · 2017
Cited alongside, same era.
Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games,
H. Owhadi, · 2017
Cited alongside, same era.
Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games,
H. Owhadi, · 2017
Cited alongside, same era.
Later among the works it cites.
Dspnet: A lightweight dilated convolution neural networks for spectral deconvolution with self-paced learning,
H. Zhu, Y. Qiao, G. Xu, L. Deng, Y. Yu-Feng, · 2019
Later among the works it cites.
The spectral bias of the deep image prior,
P. Chakrabarty, S. Maji, · 2019
Later among the works it cites.
Machine learning and computational mathematics,
E. Weinan, · 2020
Later among the works it cites.
Weak adversarial networks for high-dimensional partial differential equations,
Y. Zang, G. Bao, X. Ye, H. Zhou, · 2020
Later among the works it cites.
Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks,
Z.-Q. J. Xu, Y. Zhang, T. Luo, Y. Xiao, Z. Ma, · 2020
Later among the works it cites.
Spectrum dependent learning curves in kernel regression and wide neural networks,
B. Bordelon, A. Canatar, C. Pehlevan, · 2020
Later among the works it cites.
On the exact computation of linear frequency principle dynamics and its generalization,
T. Luo, Z. Ma, Z.-Q. J. Xu, Y. Zhang, · 2020
Later among the works it cites.
The slow deterioration of the generalization error of the random feature model,
C. Ma, L. Wu, W. E, · 2020
Later among the works it cites.
D-netpad: An explainable and interpretable iris presentation attack detector,
R. Sharma, A. Ross, · 2020
Later among the works it cites.
A phase shift deep neural network for high frequency approximation and wave problems,
W. Cai, X. Li, L. Liu, · 2020
Later among the works it cites.
Adaptive activation functions accelerate convergence in deep and physics-informed neural networks,
A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, · 2020
Later among the works it cites.
Multi-scale deep neural network (mscalednn) methods for oscillatory stokes flows in complex domains,
B. Wang, W. Zhang, W. Cai, · 2020
Later among the works it cites.
Fourier features let networks learn high frequency functions in low dimensional domains,
M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. T. Barron, R. Ng, · 2020
Later among the works it cites.
Deep residual neural networks resolve quartet molecular phylogenies.,
Z. Zou, H. Zhang, Y. Guan, J. Zhang, · 2020
Later among the works it cites.
A comparative investigation of neural networks in solving differential equations,
E. Shi, C. Xu, · 2021
Closest in time.
Generalized rough polyharmonic splines for multiscale pdes with rough coefficients,
X. Liu, L. Zhang, S. Zhu, · 2021
Closest in time.
A linear frequency principle model to understand the absence of overfitting in neural networks,
Y. Zhang, T. Luo, Z. Ma, Z.-Q. J. Xu, · 2021
Closest in time.
Deep frequency principle towards understanding why deeper learning is faster,
Z.-Q. J. Xu, H. Zhou, · 2021
Closest in time.
On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks,
S. Wang, H. Wang, P. Perdikaris, · 2021
Closest in time.