SelectNet: Self-paced learning for high-dimensional partial differential equations

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 network-based model, we introduce novel self-paced learning framework, SelectNet, which quantifies difficulty training samples, treats samples equally in early stage training, slowly explores more challenging e.g., larger residual errors, mimicking human cognitive process for efficient learning. In particular, selection network PDE solution are trained simultaneously; adaptively weighting achieving goal Numerical examples indicate that proposed SelectNet model outperforms existing models on speed robustness, especially low-regularity solutions.