论文标题
解决高维PDE的算法：从非线性蒙特卡洛到机器学习
Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
论文作者
论文摘要
近年来，使用非线性（多层次）蒙特卡洛或深度学习的思想，在很高的维度上解决了在很高维度上解决部分微分方程（PDE）的数值算法的巨大进展。对于许多不同的应用，在某些非线性蒙特卡洛方法的非线性抛物线PDES中，它们可能不受许多不同应用的维度诅咒。 在本文中，我们回顾了这些数值和理论进步。除了基于原始问题的随机重新印度的算法，例如多级PICARD迭代和Deep BSDE方法，我们还讨论了基于更传统的Ritz，Galerkin和最不正方形的配方的算法。我们希望向读者证明，在不久的将来，研究PDE以及非常高维度的控制和变异问题很可能是数学和科学计算中最有希望的新方向之一。
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.
