Space-Time Deep Neural Network Approximations for High-Dimensional Partial Differential Equations
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2025-07-15
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Journal Article
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Abstract
It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision epsilon > 0 grows exponentially in the PDE dimension and/or the reciprocal of epsilon . Recently, certain deep learning based methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep artificial neural network (ANN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating deep ANNs grows at most polynomially in both the PDE dimension d E is an element of N and the reciprocal of the prescribed approximation accuracy epsilon > 0. There are now also a few rigorous mathematical results in the scientific literature which substantiate this conjecture by proving that deep ANNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that deep ANNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point T > 0 and on a compact cube [ a, b ](d) in space but none of these results provides an answer to the question whether the entire PDE solution on [0 , T ] x [ a, b ](d )can be approximated by deep ANNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every a is an element of R , b is an element of ( a, infinity) that solutions of certain Kolmogorov PDEs can be approximated by deep ANNs on the space-time region [0 , T ] x [ a, b ](d) without the curse of dimensionality. *
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43 (4)
Pages / Article No.
918 - 975
Publisher
Global Science Press
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Subject
Deep artificial neural network; Curse of dimensionality; Approximation; Partial differential equation; PDE; Stochastic differential equation; SDE; Monte Carlo Euler; Feynman-Kac formula; ANN
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Notes
Funding
175699 - Higher order numerical approximation methods for stochastic partial differential equations (SNF)
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