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Artificial neural networks (ANNs) have very successfully been used innumerical simulations for a series of computational problems ranging from imageclassification/image recognition, speech recognition, time series analysis,game intelligence, and computational advertising to numerical approximations ofpartial differential equations (PDEs). Such numerical simulations suggest thatANNs have the capacity to very efficiently approximate high-dimensionalfunctions and, especially, indicate that ANNs seem to admit the fundamentalpower to overcome the curse of dimensionality when approximating thehigh-dimensional functions appearing in the above named computational problems.There are a series of rigorous mathematical approximation results for ANNs inthe scientific literature. Some of them prove convergence without convergencerates and some even rigorously establish convergence rates but there are only afew special cases where mathematical results can rigorously explain theempirical success of ANNs when approximating high-dimensional functions. Thekey contribution of this article is to disclose that ANNs can efficientlyapproximate high-dimensional functions in the case of numerical approximationsof Black-Scholes PDEs. More precisely, this work reveals that the number ofrequired parameters of an ANN to approximate the solution of the Black-ScholesPDE grows at most polynomially in both the reciprocal of the prescribedapproximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in\mathbb{N}$. We thereby prove, for the first time, that ANNs do indeed overcomethe curse of dimensionality in the numerical approximation of Black-ScholesPDEs.

memoirs of the american mathematical society

A proof that artificial neural networks overcome the curse of  dimensionality in the numerical approximation of Black-Scholes partial  differential equations