A Proof that Aritificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations PDF Download
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Author: Philipp Grohs
Publisher:
ISBN: 9781470474485
Category : Deep learning (Machine learning)
Languages : en
Pages : 0
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Book Description
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy e>0 and the PDE dimension deN. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.
Author: Philipp Grohs
Publisher:
ISBN: 9781470474485
Category : Deep learning (Machine learning)
Languages : en
Pages : 0
Get Book
Book Description
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy e>0 and the PDE dimension deN. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.
Author: Philipp Grohs
Publisher: American Mathematical Society
ISBN: 147045632X
Category : Mathematics
Languages : en
Pages : 106
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Author: Philipp Grohs
Publisher: Cambridge University Press
ISBN: 1316516784
Category : Computers
Languages : en
Pages : 493
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Book Description
A mathematical introduction to deep learning, written by a group of leading experts in the field.
Author: Hiroshi Kihara
Publisher: American Mathematical Society
ISBN: 1470465426
Category : Mathematics
Languages : en
Pages : 144
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Author: Björn Sandstede
Publisher: American Mathematical Society
ISBN: 1470463091
Category : Mathematics
Languages : en
Pages : 116
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Author: Rafael von Känel
Publisher: American Mathematical Society
ISBN: 1470464160
Category : Mathematics
Languages : en
Pages : 154
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Author: Olivier Bernardi
Publisher: American Mathematical Society
ISBN: 1470466996
Category : Mathematics
Languages : en
Pages : 188
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Author: David A. Craven
Publisher: American Mathematical Society
ISBN: 147046702X
Category : Mathematics
Languages : en
Pages : 226
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Author: Pablo Candela
Publisher: American Mathematical Society
ISBN: 1470465485
Category : Mathematics
Languages : en
Pages : 114
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Author: Roelof Bruggeman
Publisher: American Mathematical Society
ISBN: 1470465450
Category : Mathematics
Languages : en
Pages : 186
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