
arXiv: 2209.04997
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.
65M22, 60H15, 65C30, 68T07, high-dimensional problems, convolutional neural network, Numerical Analysis (math.NA), Neural networks for/in biological studies, artificial life and related topics, Allen-Cahn equation, Finite difference methods for initial value and initial-boundary value problems involving PDEs, Black-Scholes-Barenblatt equation, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, numerical experiments, Hamilton-Jacobi-Bellman equation, Artificial neural networks and deep learning
65M22, 60H15, 65C30, 68T07, high-dimensional problems, convolutional neural network, Numerical Analysis (math.NA), Neural networks for/in biological studies, artificial life and related topics, Allen-Cahn equation, Finite difference methods for initial value and initial-boundary value problems involving PDEs, Black-Scholes-Barenblatt equation, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs, numerical experiments, Hamilton-Jacobi-Bellman equation, Artificial neural networks and deep learning
| selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 1 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Average | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
