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We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning.
On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers
F. Brezzi · 1974
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Navier-Stokes Equations
R. Temam · 1984
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Continuous valued neural networks with two hidden layers are sufficient
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D. F. Specht · 1991
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Approximation properties of a multilayered feedforward artificial neural network
H. N. Mhaskar · 1993
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An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera · 2004
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Neural networks: a comprehensive foundation
S. Haykin · 2004
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Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, 2005
H. C. Elman, D. J. Silvester, and A. J. Wathen · 2005
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A reduced basis element method for the steady Stokes problem
A. E. Løvgren, Y. Maday, and E. M. Rønquist · 2006
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On the stability of the reduced basis method for Stokes equations in parametrized domains
G. Rozza and K. Veroy · 2007
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Learning deep architectures for ai
Y. Bengio et al · 2009
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Nonlinear model reduction via discrete empirical interpolation
S. Chaturantabut and D. C. Sorensen · 2010
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Natural language processing (almost) from scratch
R. Collobert, J. Weston, L. Bottou, M. Karlen, K. Kavukcuoglu, and P. Kuksa · 2011
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ImageNet classification with deep convolutional neural networks
A. Krizhevsky, I. Sutskever, and G. E. Hinton · 2012
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Deep neural networks for object detection
C. Szegedy, A. Toshev, and D. Erhan · 2013
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Proper orthogonal decomposition: Theory and reduced-order modelling
S. Volkwein · 2013
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Convolutional neural networks for sentence classification
Y. Kim · 2014
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Adam: A method for stochastic optimization
D. P. Kingma and J. Ba · 2014
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An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows
A. Manzoni · 2014
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A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries
A. Abdulle and O. Budáč · 2015
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Residual attention network for image classification
F. Wang, M. Jiang, C. Qian, S. Yang, C. Li, H. Zhang, X. Wang, and X. Tang · 2017
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Hyper-reduced order models for parametrized unsteady navier-stokes equations on domains with variable shape
N. Dal Santo and Manzoni · 2018
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A mean-field optimal control formulation of deep learning
W. E, J. Han, and Q. Li · 2018
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The deep ritz method: A deep learning-based numerical algorithm for solving variational problems
W. E and B. Yu · 2018
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Which neural net architectures give rise to exploding and vanishing gradients?
B. Hanin · 2018
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Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations
F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza · 2015
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Deep learning for event-driven stock prediction
X. Ding, Y. Zhang, T. Liu, and J. Duan · 2015
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Recurrent convolutional neural networks for text classification
S. Lai, L. Xu, K. Liu, and J. Zhao · 2015
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Inceptionism: Going deeper into neural networks, 2015
A. Mordvintsev, C. Olah, and M. Tyka · 2015
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Efficient model reduction of parametrized systems by matrix discrete empirical interpolation
F. Negri, A. Manzoni, and D. Amsallem · 2015
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Deep learning
I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio · 2016
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Certified reduced basis methods for parametrized partial differential equations
J. S. Hesthaven, G. Rozza, and B. Stamm · 2016
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B. Hanin and D. Rolnick · 2018
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Non-intrusive reduced order modeling of nonlinear problems using neural networks
J. S. Hesthaven and S. Ubbiali · 2018
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Deep learning for image-based cancer detection and diagnosis- a survey
Z. Hu, J. Tang, Z. Wang, K. Zhang, L. Zhang, and Q. Sun · 2018
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A comprehensive analysis of deep regression
S. Lathuilière, P. Mesejo, X. Alameda-Pineda, and R. Horaud · 2018
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Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
K. Lee and K. Carlberg · 2018
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Topological properties of the set of functions generated by neural networks of fixed size
P. Petersen, M. Raslan, and F. Voigtlaender · 2018
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Recent trends in deep learning based natural language processing
T. Young, D. Hazarika, S. Poria, and E. Cambria · 2018
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Optimal approximation with sparsely connected deep neural networks
H. Bölcskei, P. Grohs, G. Kutyniok, and P. Petersen · 2019
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An algebraic least squares reduced basis method for the solution of nonaffinely parametrized stokes equations
N. Dal Santo, S. Deparis, A. Manzoni, and A. Quarteroni · 2019
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Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in uq
C. Schwab and J. Zech · 2019
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Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem
Q. Wang, J. S. Hesthaven, and D. Ray · 2019
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