With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but 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 historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.

翻译：随着深度学习在科学计算中的最新研究，物理信息神经网络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能有效处理高维问题，但其精度相对较低，尤其对于高度不规则问题。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，以采样额外点，这些点随后与历史数据一起训练，从而加速损失收敛并获得更高精度。针对二维和十维问题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最先进的精度，同时与传统数值求解器性能相当。