Active learning based sampling for high-dimensional nonlinear partial differential equations
Active learning based sampling for high-dimensional nonlinear partial differential equations
复制标题
基于主动学习的高维非线性偏微分方程采样
DOI：
10.1016/j.jcp.2022.111848
复制
发表时间：
2023
影响因子：
4.1
通讯作者：
Wang, Chunmei
中科院分区：
文献类型：
--
作者：
Gao, Wenhan;Wang, Chunmei
基于深度学习的最小二乘法在解决高维和非线性偏微分方程（PDE）方面取得了成功的结果。然而，这种方法通常收敛缓慢。为了加快该方法的收敛速度，本文提出了一种基于主动学习的采样算法。该算法基于残差从概率密度函数中主动选择信息量最大的训练样本，以减少误差。特别是，残留误差较大的点将有更多机会被选择进行训练。这种算法模仿了人类的学习过程：学习者可能会花更多的时间重复学习错误，而不是他们正确完成的其他任务。一系列的数值结果表明，我们的主动学习为基础的采样在高维加速基于深度学习的最小二乘方法的收敛的有效性。
The deep-learning-based least squares method has shown successful results in solving high-dimensional and non-linear partial differential equations (PDEs). However, this method usually converges slowly. To speed up the convergence of this approach, an active-learning-based sampling algorithm is proposed in this paper. This algorithm actively chooses the most informative training samples from a probability density function based on residual errors to facilitate error reduction. In particular, points with larger residual errors will have more chances of being selected for training. This algorithm imitates the human learning process: learners are likely to spend more time repeatedly studying mistakes than other tasks they have correctly finished. A series of numerical results are illustrated to demonstrate the effectiveness of our active-learning-based sampling in high dimensions to speed up the convergence of the deep-learning-based least squares method.