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

Hornik, Kurt; Stinchcombe, Maxwell B.; White, Halbert

doi:10.1016/0893-6080(90)90005-6

Cited by 1,782 publications

(827 citation statements)

References 4 publications

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“…The approximation theorems of Ref. [25] and well known time-series embedding results Ref. [41] together establish an equivalence (given properly…”

Section: Constructionsupporting

confidence: 54%

“…Setting the issues of a finite number of parameters aside and leaving any constraints on the weights behind, the neural networks we utilize can approximateF and its derivatives (to any order) to arbitrary accuracy [24] [25]. Precise statements of the neural network approximation theorems require machinery from functional analysis (see [2]) and is covered in detail in the papers by Hornik et.…”

Section: Constructionmentioning

confidence: 99%

“…From the functional analysis perspective, the neural networks we study are diverse universal function approximators -they can approximate nearly any mapping one might wish to approximate [25]. Thus, studying the space of neural networks represents a practical study of a common space of mappings used by time-series analysists to reconstruct unknown dynamics from time-series data [17] [40] [27].…”

Section: B Why Scalar Neural Network?mentioning

confidence: 99%

See 2 more Smart Citations

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Albers

Sprott

2006

J Stat Phys

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Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.

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“…The approximation theorems of Ref. [25] and well known time-series embedding results Ref. [41] together establish an equivalence (given properly…”

Section: Constructionsupporting

confidence: 54%

Section: Constructionmentioning

confidence: 99%

Section: B Why Scalar Neural Network?mentioning

confidence: 99%

See 1 more Smart Citation

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Albers

Sprott

2006

J Stat Phys

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“…any C r mapping and its derivatives to arbitrary accuracy, given enough neurons). That scalar neural networks can approximate the mappings we are interested in is a topic addressed in [21]. Combining the approximation theorems of Hornik et al [21] and the time-series embedding results of Takens [36] shows the equivalence between our neural networks and the dynamical systems from compact sets in R n to themselves (for specific arguments along these lines, see [1]).…”

mentioning

confidence: 99%

Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

Albers

Sprott

2006

Physica D: Nonlinear Phenomena

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This paper examines the most probable route to chaos in high-dimensional dynamical systems in a very general computational setting. The most probable route to chaos in high-dimensional, discrete-time maps is observed to be a sequence of Neimark-Sacker bifurcations into chaos. A means for determining and understanding the degree to which the Landau-Hopf route to turbulence is non-generic in the space of C r mappings is outlined. The results comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. In their first edition of Fluid Mechanics [26], Landau and Lifschitz proposed a route to turbulence in fluid systems. Since then, much work, in dynamical systems, experimental fluid dynamics, and many other fields has been done concerning the routes to turbulence. In this paper, we present early results from the first large statistical study of the route to chaos in a very general class of high-dimensional, C r , dynamical systems. Our results contain both some reassurances based on a wealth of previous results and some surprises. We conclude that, for high-dimensional discrete-time maps, the most probable route to chaos (in our general construction) from a fixed point is via at least one Neimark-Sacker bifurcation, followed by persistent zero Lyapunov exponents, and finally a bifurcation into chaos. We observe both the Ruelle-Takens

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“…We do so because Hornik, Stinchcombe and White (1990) showed that a function and its derivatives of any unknown functional form can be approximated arbitrarily accurately by such a neural network. Specifically, after estimating the derivatives in (10) with a neural network, we estimate the Jacobian .…”

Section: The Music Lab Experiments and The Matthew Mechanismmentioning

confidence: 99%

Social influence and the Matthew mechanism: The case of an artificial cultural market

Bask

2014

Physica A: Statistical Mechanics and its Applications

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. We show that the Matthew effect, or Matthew mechanism, was present in the artificial cultural market Music Lab when social influence between individuals was allowed, whereas this was not the case when social influence was not allowed. We also sketch on a class of social network models, derived from social influence theory, that may generate the Matthew effect. Thus, we propose a theoretical framework that may explain why the most popular songs were much more popular, and the least popular songs were much less popular, than when disallowing social influence between individuals. Terms of use: Documents inJEL codes: C31, C65, Z19.

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

Cited by 1,782 publications

References 4 publications

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

Social influence and the Matthew mechanism: The case of an artificial cultural market

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