We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only based ASM algorithm denoted by ASM I. In this approach, we first train the neural network by using a small number of residual points and divide the computational domain into a certain number of sub-domains, we then add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have largest absolute values of the residual in this sub-domain will be added as new residual points. We further develop a second type of ASM algorithm (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may be not able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I except that in ASM II, we add new residual points which not only have large residual but also large gradient. To demonstrate the effectiveness of the present methods, we employ both ASM I and ASM II to solve a number of PDEs, including Burger equation, compressible Euler equation, Poisson equation over an L-shape domain as well as high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II algorithm, and both methods deliver much more accurate solution than original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency and stability compared with the ASM I algorithm.

翻译：本文考虑采用物理信息神经网络(PINNs)求解具有尖锐解的正向与逆向偏微分方程。为更好地捕捉解的尖锐特征，我们提出基于残差和解梯度的自适应采样方法(ASMs)。首先给出仅基于残差的ASM算法(记为ASM I)：该方案先利用少量残差点训练神经网络并将计算域划分为若干子域，随后在残差绝对值均值最大的子域内添加新的残差点——选取该子域中残差绝对值最大的点作为新增残差点。鉴于仅依赖残差可能无法有效捕捉解的尖锐性，我们进一步开发了基于残差和解梯度的第二种ASM算法(记为ASM II)。ASM II的流程与ASM I基本一致，区别在于ASM II新增的残差点需同时具有较大的残差和梯度。为验证所提方法的有效性，我们分别采用ASM I和ASM II求解了包含Burgers方程、可压缩Euler方程、L形域Poisson方程及高维Poisson方程在内的多个偏微分方程。数值结果表明，ASM I和ASM II算法均能良好逼近尖锐解，且在相同残差点数量下，两种方法均比原始PINNs获得更精确的解。此外，相比ASM I算法，ASM II算法在精度、效率和稳定性方面表现更优。