论文标题
强大的基于SDE的变异配方，用于通过深度学习求解线性PDE
Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
论文作者
论文摘要
蒙特卡洛方法和深度学习的组合最近导致了在高维度中求解部分微分方程（PDE）的有效算法。相关的学习问题通常被称为基于相关随机微分方程（SDE）的变异公式，可以使用基于梯度的优化方法最小化相应损失。因此，在各自的数值实现中，至关重要的是要依靠足够的梯度估计器，这些梯度估计器表现出较低的差异，以便准确，迅速地达到收敛性。在本文中，我们严格研究了在线性kolmogorov pdes的上下文中出现的相应数值方面。特别是，我们系统地比较了现有的深度学习方法并为其表演提供理论解释。随后，我们建议的新方法在理论上和数字上都可以证明更健壮，从而导致了实质性的改进。
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements.
