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

偏微分方程（PDE）是对许多真实的世界现象进行建模的关键工具。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，并且不能显式求解。这是一个非常具有挑战性的任务，可证明解决这种高维非线性偏微分方程近似没有遭受所谓的灾难的维数。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。特别是，模拟表明，基于DNN的算法克服了某些非线性偏微分方程的解的数值近似的维数灾难。对于某些线性和非线性偏微分方程，这也已在数学上得到证明。该项目的主要目标是首次严格证明DNN克服了一类随机控制问题产生的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灾难。

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.