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.
A numerical algorithm for a class of BSDEs via branching process
Société Générale, Global Market Quantitative Research（法国兴业银行全球市场量化研究部）；CEREMADE, University of Paris-Dauphine（巴黎第九大学 CEREMADE 中心）；Ecole Polytechnique Paris, Centre de Mathématiques Appliquées（巴黎综合理工学院应用数学中心）
The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions
提出了一种新的离散化方法，可同时求解带可微系数的正倒向随机微分方程（FBS-R）及其Malliavin灵敏度问题.控制过程估计由相应的线性Bracket驱动的Malliavin衍生物的解决方案对，这意味着需要提供准确的$Γ$估计的轨迹。近似是基于Feynman-Kac公式和Malliavin链式法则给出的合并公式。连续时间动态离散的θ-计划。为了允许一个有效的数值解所产生的半离散条件期望在可能的高维，它是基本的，所选择的方法承认可微的估计。考虑了两种完全可实现的方案：BCOS方法作为一维框架中的参考，以及高维问题中的神经网络蒙特卡洛回归，类似于最近出现的一类深度BCOS方法[Han et al.（2018），Huré et al.（2020）]。误差分析表明，在标准Lipschitz假设和前向扩散中的加性噪声下，$L^2$收敛于$1/2$阶。数值实验提供了一系列不同的半线性和准线性方程的$50$维，表明所提出的计划产生了显着的改善控制估计。
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
Beijing Institute of Big Data Research (China)（北京大数据研究院（中国））；Princeton University (USA)（普林斯顿大学（美国））；Peking University (China)（北京大学（中国））；ETH Zurich (Switzerland)（苏黎世联邦理工学院（瑞士））
Deep Truncated FBSDE Method for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs
Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）；Department of Mathematics, Friedrich-Alexander-Universität Erlangen–Nürnberg（埃尔朗根-纽伦堡大学数学系）
Averaged Adam accelerates stochastic optimization in the training of deep neural network approximations for partial differential equation and optimal control problems
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
本文受 Polyak-Ruppert 平均化启发，将平均化 Adam 用于 PDE、最优控制及图像分类的 DNN 训练，并在 PINN、deep BSDE 等多个数值实验中证明其优于标准 Adam 与 SGD。
