论文标题
基于高维完全非线性合并PDE和2BSDE的深卷积神经网络的数值近似
Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs
论文作者
论文摘要
本文提出了两种有效的近似方法，以解决高维完全非线性偏微分方程（NPDES）和二阶向后随机微分方程（2BSDE），因为这样的高维度完全NPDE很难使标准近似方法的计算成本相关，而标准近似方法的分解也会成倍增长，而差异呈上型。因此，我们考虑以下方法来克服这一困难。对于合并的完全NPDES和2BSDES系统，结合时间远期离散化和Relu功能，我们使用多尺度的深度学习融合和卷积神经网络（CNN）技术分别获得两个数值近似方案。最后，给出了三个实用的高维测试问题，涉及艾伦·卡恩，黑人choles-barentblatt和hamiltonian-jacobi-bellman方程式，以便与现有方法相比，第一个提出的方法表现出更高的效率和准确性，而第二个提出的方法可以超过400 $ dimersions的效率超过400 $ dimersions，该方法扩展了dimensions the Inmully的效率。
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 multi-scale 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-Barentblatt, and Hamiltonian-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.
