论文标题
SelectNet：高维偏微分方程的自定进度学习
SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations
论文作者
论文摘要
具有深层神经网络作为功能参数化的最小二乘法已应用于成功求解某些高维偏微分方程（PDE）。但是，它的收敛速度很慢，即使在简单的PDE中，也可能无法保证。为了改善基于网络的最小二乘模型的收敛性，我们介绍了一个新颖的自定进度学习框架Selectnet，该框架量化了训练样本的难度，在训练的早期阶段同样对待样本，并慢慢探索更具挑战性的样本，例如，样本具有更大的残留错误，模仿了人类认知过程以获得更有效的学习过程。特别是，选择网络和PDE解决方案网络同时训练；选择网络可适应解决方案网络的训练样本，从而实现了自定进度学习的目标。数值示例表明，所提出的SelectNet模型在收敛速度和收敛鲁棒性上的表现优于现有模型，尤其是对于低规度解决方案。
The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. To improve the convergence of the network-based least squares model, we introduce a novel self-paced learning framework, SelectNet, which quantifies the difficulty of training samples, treats samples equally in the early stage of training, and slowly explores more challenging samples, e.g., samples with larger residual errors, mimicking the human cognitive process for more efficient learning. In particular, a selection network and the PDE solution network are trained simultaneously; the selection network adaptively weighting the training samples of the solution network achieving the goal of self-paced learning. Numerical examples indicate that the proposed SelectNet model outperforms existing models on the convergence speed and the convergence robustness, especially for low-regularity solutions.
