With recent study of the deep learning in scientific computation, the PINNs method has drawn widespread attention for solving PDEs. Compared with traditional methods, PINNs can efficiently handle high-dimensional problems, while the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with history data to speed up the convergence of loss and achieve a higher accuracy. Several numerical simulations on 2d to 10d problems show that GAS is a promising method which achieves the state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.

翻译：摘要：随着深度学习在科学计算领域的近期研究，PINNs方法在求解偏微分方程方面引起了广泛关注。与传统方法相比，PINNs能够高效处理高维问题，但其精度相对较低，尤其对于高度不规则的问题。受自适应有限元方法和增量学习思想的启发，我们提出了GAS，一种基于高斯混合分布的自适应采样方法用于PINNs。在训练过程中，GAS利用当前残差信息生成高斯混合分布，用于附加点的采样，然后这些附加点与历史数据一起训练，以加速损失收敛并获得更高精度。在2维至10维问题上的多个数值模拟表明，GAS是一种有前景的方法，在深度求解器中达到了当前最优精度，同时与传统数值求解器性能相当。