Deep neural networks overcome the curse of dimensionality in the numerical approximation of stochastic control problems and of semilinear Poisson equations

深度神经网络克服了随机控制问题和半线性泊松方程数值逼近中的维数灾难

• 批准号: 464101154
• 负责人: Professor Dr. Martin Hutzenthaler
• 金额: --
• 依托单位: Arbeitsgruppe Numerische Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/464101154?language=en
• 关键词: Deep; neural; networks; overcome; curse

Partial differential equations (PDEs) are a key tool in the modeling of many real world phenomena. Several PDEs that arise in financial engineering, economics, quantum mechanics or statistical physics are nonlinear, high-dimensional, and cannot be solved explicitly. It is a highly challenging task to provably solve such high-dimensional nonlinear PDEs approximately without suffering from the so-called curse of dimensionality. Deep neural networks (DNNs) and other deep learning-based methods have recently been applied very successfully to a number of computational problems. In particular, simulations indicate that algorithms based on DNNs overcome the curse of dimensionality in the numerical approximation of solutions of certain nonlinear PDEs. For certain linear and nonlinear PDEs this has also been proven mathematically. The key goal of this project is to rigorously prove for the first time that DNNs overcome the curse of dimensionality for a class of nonlinear PDEs arising from stochastic control problems and for a class of semilinear Poisson equations with Dirichlet boundary conditions.

偏微分方程（PDEs）是许多现实世界现象建模的关键工具。在金融工程、经济学、量子力学或统计物理学中出现的一些偏微分方程是非线性的、高维的，并且不能明确地求解。如何在不遭受所谓的维数诅咒的情况下，证明地近似求解这种高维非线性偏微分方程是一项极具挑战性的任务。深度神经网络（dnn）和其他基于深度学习的方法最近已经非常成功地应用于许多计算问题。具体而言，仿真结果表明，基于深度神经网络的算法克服了某些非线性偏微分方程数值逼近解的维数问题。对于某些线性和非线性偏微分方程，也用数学方法证明了这一点。该项目的主要目标是首次严格证明dnn克服了由随机控制问题引起的一类非线性偏微分方程和一类具有Dirichlet边界条件的半线性泊松方程的维数诅咒。