Numerical approximation based on deep convolutional neural network for high‐dimensional fully nonlinear merged PDEs and 2BSDEs

This paper proposes two efficient approximation methods to solve high‐dimensional fully nonlinear partial differential equations (NPDEs) and second‐order backward stochastic differential equations (2BSDEs), where such high‐dimensional fully NPDEs are extremely difficult to solve because the computat...

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Vydané v:	Mathematical methods in the applied sciences Ročník 47; číslo 7; s. 6184 - 6204
Hlavní autori:	Xiao, Xu, Qiu, Wenlin, Nikan, Omid
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Freiburg Wiley Subscription Services, Inc 15.05.2024
Predmet:	Allen–Cahn equation Approximation Artificial neural networks Black–Scholes–Barenblatt equation Computational efficiency convolutional neural network Hamilton–Jacobi–Bellman equation high‐dimensional problems Machine learning Neural networks Nonlinear differential equations numerical experiments Partial differential equations ReLU second‐order backward stochastic differential equations
ISSN:	0170-4214, 1099-1476
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Popis
Shrnutí:	This paper proposes two efficient approximation methods to solve high‐dimensional fully nonlinear partial differential equations (NPDEs) and second‐order backward stochastic differential equations (2BSDEs), where such high‐dimensional fully NPDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully NPDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multiscale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. Finally, three practical high‐dimensional test problems involving Allen–Cahn, Black–Scholes–Barenblatt, and Hamilton–Jacobi–Bellman equations are given so that the first proposed method exhibits higher efficiency and accuracy than the existing method, while the second proposed method can extend the dimensionality of the completely NPDEs–2BSDEs system over 400 dimensions, from which the numerical results highlight the effectiveness of proposed methods.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0170-4214 1099-1476
DOI:	10.1002/mma.9915