Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks

By Kurt Hornik, Maxwell B. Stinchcombe, Halbert White. 1990

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Citations in the corpus

Meta-Learning and Universality: Deep Representations and Gradient Descent can Approximate any Learning Algorithm (2017, methodology): “Furthermore, Hornik et al. (1990) showed that a neural network with a single hidden layer can simultaneously approximate any function and its derivatives, under mild assumptions on the activation function used and target function’s domain.”
Graph Embedding Techniques, Applications, and Performance: A Survey (2017, background): “Deep learning methods can model a wide range of functions following the universal approximation theorem [36]: given enough parameters, they can learn the mix of community and structural equivalence, to embed the nodes such that the reconstruction error is minimized.”
DGM: A deep learning algorithm for solving partial differential equations (2017, neutral): “Inregards to general results on the approximation power of neural networks we refer the interested reader to classical works [11, 25, 26, 39] and we also mention the recent work by [38], where the authors study the necessary and suﬃcient complexity of ReLU neural networks that is required for…”
Drought forecasting using artificial neural networks and time series of drought indices (2007, background): “Given sufficient data and complexity, ANN can be designed to model any relationship between a series of independent and dependent variables – inputs and outputs to the network respectively (Hornik et al., 1990; Luk et al., 2000).”
A unified deep artificial neural network approach to partial differential equations in complex geometries (2017, background): “Single hidden layer networks are sufficient to approximate any continuous function to any accuracy by having a sufficient amount of neurons in the hidden layer [12, 13, 24].”
Optimal Adaptive Control and Differential Games by Reinforcement Learning Principles (2012, methodology): “In the neural-network research, approximation results are shown for other basis sets including sigmoid, hyperbolic tangent, Gaussian radial basis functions and others (Hornik et al., 1990; Sandberg, 1998).”
Predicting breast cancer survivability: a comparison of three data mining methods (2005, background): “[31] empirically showed that given the right size and the structure, MLP is capable of learning arbitrarily complex nonlinear functions to arbitrary accuracy levels.”
Learning in Artificial Neural Networks: A Statistical Perspective (1989, background): “…as a single hidden layer are capable of arbitrarily accurate approximation to an arbitrary mapping provided that sufficiently many hidden units are available [see Carroll and Dick-inson 1989; Cybenko 1989; Funahashi 1989; Hecht-Nielsen 1989; Hornik et al. 1989a,b (HSW); Stinchcombe and White 19891.”
Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism (2020, background): “merged as a very promising application of deep neural networks in engineering and science [1], as part of the broader ﬁeld of scientiﬁc machine learning [2]. PINNs are powerful function approximators [3, 4], made possible by recent advances in GPU capabilities and training algorithms for deep neural”
Optimal tracking control of nonlinear partially-unknown constrained-input systems using integral reinforcement learning (2014, methodology): “They used an iterative online policy iteration (PI) (Howard, 1960) procedure to implement their IRL algorithm. Later, inspired by Vrabie and Lewis (2009) and Vrabie et al. (2009), some online PI algorithms were presented to solve the optimal regulation problem for completely unknown linear systems (”
Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability (2001, background): “(1989) showed that the output function does not have to be continuous, and they also proved that a neural network can approximate simultaneously both a function and its derivative (Hornik et al. 1990).”