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Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions.
On Stirling’s Theorem as a Definition of the Gamma Function
Egan, M. F · 1933
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A remark on Stirling’s formula
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Approximation by superpositions of a sigmoidal function
Cybenko, G · 1989
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On the approximate realization of continuous mappings by neural networks
Funahashi, K.-I · 1989
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Multilayer feedforward networks are universal approximators
Hornik, K., Stinchcombe, M., and White, H · 1989
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Approximation capabilities of multilayer feedforward networks
Hornik, K · 1991
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Neural net approximation
Barron, A · 1992
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A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training
Jones, L. K · 1992
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Universal Approximation Bounds for Superpositions of a Sigmoidal Function
Barron, A · 1993
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Rates of convergence for radial basis functions and neural networks
Girosi, F., and Anzellotti, G · 1993
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Multilayer Feedforward Networks With a Nonpolynomial Activation Function Can Approximate Any Function
Leshno, M., Lin, V. Y., Pinkus, A., and Schocken, S · 1993
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Approximation and estimation bounds for artificial neural networks
Barron, A · 1994
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Rates of convex approximation in non-Hilbert spaces
Donahue, M. J., Darken, C., Gurvits, L., and Sontag, E · 1997
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Approximation and Learning of Convex Superpositions
Gurvits, L., and Koiran, P · 1997
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Estimates of the Number of Hidden Units and Variation with Respect to Half-Spaces
Kůrková, V., Kainen, P. C., and Kreinovich, V · 1997
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Lower bounds for approximation by MLP neural networks
Maiorov, V., and Pinkus, A · 1999
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Approximation theory of the MLP model in neural networks
Pinkus, A · 1999
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Comparison of worst case errors in linear and neural network approximation
Kůrková, V., and Sanguineti, M · 2002
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Probability theory
Klenke, A · 2006
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Minimization of Error Functionals over Perceptron Networks
Kůrková, V · 2008
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Geometric Upper Bounds on Rates of Variable-Basis Approximation
Kůrková, V., and Sanguineti, M · 2008
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Complexity of Gaussian-radial-basis networks approximating smooth functions
Kainen, P. C., Kůrková, V., and Sanguineti, M · 2009
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Dependence of Computational Models on Input Dimension: Tractability of Approximation and Optimization Tasks
Kainen, P. C., Kůrková, V., and Sanguineti, M · 2012
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The Power of Depth for Feedforward Neural Networks
Eldan, R., and Shamir, O · 2016
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Optimal approximation with sparsely connected deep neural networks
Bölcskei, H., Grohs, P., Kutyniok, G., and Petersen, P · 2019
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Uniform error estimates for artificial neural network approximations for heat equations
Gonon, L., Grohs, P., Jentzen, A., Kofler, D., and Šiška, D · 2019
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Space-time error estimates for deep neural network approximations for differential equations
Grohs, P., Hornung, F., Jentzen, A., and Zimmermann, P · 2019
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Deep neural network approximations for Monte Carlo algorithms
Grohs, P., Jentzen, A., and Salimova, D · 2019
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A theoretical analysis of deep neural networks and parametric PDEs
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Depth Separation for Neural Networks
Daniely, A · 2017
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Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks
Safran, I., and Shamir, O · 2017
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Error bounds for approximations with deep ReLU networks
Yarotsky, D · 2017
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DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
Elbrächter, D., Grohs, P., Jentzen, A., and Schwab, C · 2018
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Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P · 2018
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Approximation capability of two hidden layer feedforward neural networks with fixed weights
Guliyev, N. J., and Ismailov, V. E · 2018
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Kutyniok, G., Petersen, P., Raslan, M., and Schneider, R · 2019
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Li, B., Tang, S., and Yu, H · 2019
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Reisinger, C., and Zhang, Y · 2019
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An overview on deep learning-based approximation methods for partial differential equations
Beck, C., Hutzenthaler, M., Jentzen, A., and Kuckuck, B · 2020
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Beneventano, P., Cheridito, P., Jentzen, A., and von Wurstemberger, P · 2020
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Analysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Berner, J., Grohs, P., and Jentzen, A · 2020
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Efficient approximation of high-dimensional functions with deep neural networks
Cheridito, P., Jentzen, A., and Rossmannek, F · 2020
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Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
E, W., Han, J., and Jentzen, A · 2020
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Deep Neural Network Approximation Theory
Elbrächter, D., Perekrestenko, D., Grohs, P., and Bölcskei, H · 2020
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Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models
Gonon, L., and Schwab, C · 2020
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Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
Grohs, P., and Herrmann, L · 2020
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Space-time deep neural network approximations for high-dimensional partial differential equations
Hornung, F., Jentzen, A., and Salimova, D · 2020
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A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Hutzenthaler, M., Jentzen, A., Kruse, T., and Nguyen, T. A · 2020
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