The deep-learning-based least squares method has shown successful results in solving high-dimensional 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.
翻译:基于深层学习的最小方块方法在解决高维非线性部分差异方程式(PDEs)方面显示出成功的结果。 但是,这种方法通常会缓慢地趋同。 为加快这一方法的趋同,本文件建议采用基于积极学习的抽样算法。这一算法根据残余错误从概率密度函数中积极选择信息最丰富的培训样本,以便于减少错误。特别是,有较大残余错误的点将有更多机会被选定参加培训。这种算法仿照了人类学习过程:学习者可能比他们正确完成的其他任务多花更多时间反复学习错误。一系列数字结果展示了我们基于积极学习的高层面抽样的有效性,以加快基于深学习的最小方块方法的趋同。