论文标题
高维随机偏微分方程的预测 - 校准深度学习算法
A predictor-corrector deep learning algorithm for high dimensional stochastic partial differential equations
论文作者
论文摘要
在本文中，我们提出了一种基于深度学习的数值方法，用于近似高维的随机部分微分方程（SPDE）。在每个时间步骤中，我们的方法都依赖于预测器 - 校正程序。更确切地说，我们将原始SPDE分解为退化的SPDE和确定性的PDE。然后，在预测步骤中，我们使用Euler方案求解了退化SPDE，而在校正步骤中，我们通过其等效的向后随机微分方程（BSDE）通过深神经网络解决了二阶确定性PDE。在标准假设下，提出了误差估计和提出算法的收敛速率。数值示例说明了所提出算法的效率和准确性。
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely, we decompose the original SPDE into a degenerate SPDE and a deterministic PDE. Then in the prediction step, we solve the degenerate SPDE with the Euler scheme, while in the correction step we solve the second-order deterministic PDE by deep neural networks via its equivalent backward stochastic differential equation (BSDE). Under standard assumptions, error estimates and the rate of convergence of the proposed algorithm are presented. The efficiency and accuracy of the proposed algorithm are illustrated by numerical examples.
