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

TL;DR

This paper introduces two deep learning-based methods using CNN and multi-scale fusion to efficiently approximate solutions to high-dimensional fully nonlinear PDEs and 2BSDEs, overcoming the curse of dimensionality.

Contribution

The paper develops two novel deep learning schemes combining CNN and multi-scale techniques for high-dimensional NPDEs and 2BSDEs, achieving higher efficiency and extending dimensionality limits.

Findings

The first method outperforms existing approaches in accuracy and efficiency.

The second method successfully handles systems over 400 dimensions.

Numerical tests on Allen-Cahn, Black-Scholes, and HJB equations validate effectiveness.

Abstract

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.