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It is known that any continuous multivariate function can be represented exactly by a composition functions of a single variable - the so-called Kolmogorov-Arnold representation.
Angenäherte auflösung von systemen linearer gleichungen
S. Kaczmarz · 1937
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On functions of three variables
V. I. Arnold · 1957
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On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition
A. N. Kolmogorov · 1957
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Metric entropy, widths, and superpositions of functions
G. G. Lorentz · 1962
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On the structure of continuous functions of several variables
D. A. Sprecher · 1965
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An improvement in the smoothness of the functions in A.N. Kolmogorov’s theorem on superpositions
B. L. Fridman · 1967
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Projection methods for solving sparse linear systems
R. P. Tewarson · 1969
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An improvement in the superposition theorem of Kolmogorov
D. A. Sprecher · 1972
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Solution of underdetermined nonlinear equations by stationary iteration methods
K.-H. Meyn · 1983
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Solving systems of nonlinear equations by means of an accelerated successive orthogonal projections method
J. M. Martinez · 1986
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Parallel and sequential Kaczmarz methods for solving underdetermined nonlinear equations
J. M. Martinez and R. J. B. de Sampaio · 1986
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Kolmogorov’s mapping neural network existence theorem
R. Hecht-Nielsen · 1987
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Generalized additive models
T. J. Hastie and R. J. Tibshirani · 1990
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Multi-layer perceptrons with B-spline receptive field functions
S. Lane, M. Flax, D. Handelman, and J. Gelfand · 1990
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Kolmogorov’s theorem and multilayer neural networks
V. Kůrková · 1992
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Approximative versions of Kolmogorov’s superposition theorem, proved constructively
M. Nees · 1994
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A numerical implementation of Kolmogorov’s superpositions
D. A. Sprecher · 1996
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A numerical implementation of Kolmogorov’s superpositions II
D. A. Sprecher · 1997
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On the training of a Kolmogorov network
M. Köppen · 2002
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Kolmogorov’s spline network
B. Igelnik and N. Parikh · 2003
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On Kolmogorov’s representation of functions of several variables by functions of one variable
M. Coppejans · 2004
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Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
M. Raissi, P. Perdikaris, and G. E. Karniadakis · 2019
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Error bounds for deep ReLU networks using the Kolmogorov-Arnold superposition theorem
H. Montanelli and H. Yang · 2020
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Modelling non-linear control systems using the discrete Urysohn operator
M. Poluektov and A. Polar · 2020
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Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
M. Raissi, A. Yazdani, and G. E. Karniadakis · 2020
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Convergence rates for the stochastic gradient descent method for non-convex objective functions
B. Fehrman, B. Gess, and A. Jentzen · 2020
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Analysis of Kolmogorov’s superposition theorem and its implementation in applications with low and high dimensional data
D. Bryant · 2008
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On a constructive proof of Kolmogorov’s superposition theorem
J. Braun and M. Griebel · 2009
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Isogeometric Analysis: Toward Integration of CAD and FEA
J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs · 2009
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Nouvelles méthodes de traitement de signaux multidimensionnels par décomposition suivant le théorème de Superposition de Kolmogorov
P.-E. Leni · 2010
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A review of some results on ridge function approximation
V. E. Ismailov · 2013
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Kolmogorov superposition theorem and its applications
X. Liu · 2015
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Physics-inspired structural representations for molecules and materials
F. Musil, A. Grisafi, A. P. Bartók, C. Ortner, G. Csányi, and M. Ceriotti · 2021
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The Kolmogorov-Arnold representation theorem revisited
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A deep machine learning algorithm for construction of the Kolmogorov-Arnold representation
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Physics-informed machine learning
G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang · 2021
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KASAM: Spline additive models for function approximation
H. van Deventer, P. J. van Rensburg, and A. Bosman · 2022
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On the Kolmogorov neural networks
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KAN: Kolmogorov-Arnold Networks
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Utilizing the Kolmogorov-Arnold networks for chiller energy consumption prediction in commercial building
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A benchmarking study of Kolmogorov-Arnold networks on tabular data
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