Fetching the paper…
Reading the bibliography…
We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation.
A Jordan curve of positive area
W. Osgood · 1903
Earlier work this paper cites.
A bang-bang theorem with bounds on the number of switchings
H. Sussmann · 1979
Earlier work this paper cites.
Exact controllability, stabilization and perturbations for distributed systems
J.-L. Lions · 1988
Earlier work this paper cites.
Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
Earlier work this paper cites.
Multilayer feedforward networks are universal approximators
K. Hornik, M. Stinchcombe, H. White, et al · 1989
Earlier work this paper cites.
Multilayer feedforward networks with a nonpolynomial activation function can approximate any function
M. Leshno, V. Lin, A. Pinkus, and S. Schocken · 1993
Earlier work this paper cites.
Wavelet methods for pointwise regularity and local oscillations of functions
S. Jaffard and Y. Meyer · 1996
Earlier work this paper cites.
Complete controllability of continuous-time recurrent neural networks
E. Sontag and H. Sussmann · 1997
Earlier work this paper cites.
The optimal time-continuous mass transport problem and its augmented Lagrangian numerical resolution
J.-D. Benamou and Y. Brenier · 1998
Earlier work this paper cites.
The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
M. Shishikura · 1998
Earlier work this paper cites.
Approximation theory of the mlp model in neural networks
A. Pinkus · 1999
Earlier work this paper cites.
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
J.-D. Benamou and Y. Brenier · 2000
Earlier work this paper cites.
Simultaneous exact controllability and some applications
M. Tucsnak and G. Weiss · 2000
Earlier work this paper cites.
Error bounds for approximation with neural networks
M. Burger and A. Neubauer · 2001
Earlier work this paper cites.
Topics in optimal transportation
C. Villani · 2003
Earlier work this paper cites.
Fractal geometry: mathematical foundations and applications
K. Falconer · 2004
Cited alongside, same era.
Optimal transport: old and new
C. Villani · 2008
Cited alongside, same era.
Deep learning
Y. LeCun, Y. Bengio, and G. Hinton · 2015
Cited alongside, same era.
Control to Flocking of the Kinetic Cucker–Smale Model
B. Piccoli, F. Rossi, and E. Trélat · 2015
Cited alongside, same era.
Optimal shape and location of sensors for parabolic equations with random initial data
Y. Privat, E. Trélat, and E. Zuazua · 2015
Cited alongside, same era.
Deep residual learning for image recognition
K. He, X. Zhang, S. Ren, and J. Sun · 2016
Cited alongside, same era.
From averaged to simultaneous controllability
Nonlinear approximation and (deep) relu networks
I. Daubechies, R. DeVore, S. Foucart, B. Hanin, and G. Petrova · 2019
Later among the works it cites.
Approximate and exact controllability of the continuity equation with a localized vector field
M. Duprez, M. Morancey, and F. Rossi · 2019
Later among the works it cites.
Deep learning via dynamical systems: An approximation perspective
Q. Li, T. Lin, and Z. Shen · 2019
Later among the works it cites.
Computational optimal Transport: With Applications to Data Science
G. Peyré, M. Cuturi, et al · 2019
Later among the works it cites.
Control on the manifolds of mappings as a setting for deep learning
A. Agrachev and A. Sarychev · 2020
Later among the works it cites.
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
J. Lohéac and E. Zuazua · 2016
Cited alongside, same era.
Optimal observability of the multi-dimensional wave and schrödinger equations in quantum ergodic domains
Y. Privat, E. Trélat, and E. Zuazua · 2016
Cited alongside, same era.
Fractals in probability and analysis
C. Bishop and Y. Peres · 2017
Cited alongside, same era.
Matrix optimal mass transport: a quantum mechanical approach
Y. Chen, T. Georgiou, and A. Tannenbaum · 2017
Cited alongside, same era.
Stable architectures for deep neural networks
E. Haber and L. Ruthotto · 2017
Cited alongside, same era.
A proposal on machine learning via dynamical systems
E. Weinan · 2017
Cited alongside, same era.
Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss
L. Chizat and F. Bach · 2020
Later among the works it cites.
Deep neural networks, generic universal interpolation, and controlled ODEs
C. Cuchiero, M. Larsson, and J. Teichmann · 2020
Later among the works it cites.
Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field
M. Duprez, M. Morancey, and F. Rossi · 2020
Later among the works it cites.
Large-time asymptotics in deep learning
C. Esteve, B. Geshkovski, D. Pighin, and E. Zuazua · 2020
Later among the works it cites.
Error bounds for approximations with deep ReLU neural networks in W s , p W^{s,p} norms
I. Gühring, G. Kutyniok, and P. Petersen · 2020
Later among the works it cites.
Universal approximation power of deep neural networks via nonlinear control theory
P. Tabuada and B. Gharesifard · 2020
Later among the works it cites.
Universal approximation property of neural ordinary differential equations
T. Teshima, K. Tojo, M. Ikeda, I. Ishikawa, and K. Oono · 2020
Later among the works it cites.
Momentum residual neural networks
M. Sander, P. Ablin, M. Blondel, and G. Peyré · 2021
Closest in time.
Sparse approximation in learning via neural odes
C. E. Yagüe and B. Geshkovski · 2021
Closest in time.