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:我们考虑用物理知情神经网络(PINNs)解决前方和反面PDEs,这些前方和反面PDEs在这项工作中具有尖锐的解决方案。特别是,为了更好地捕捉解决方案的精度,我们根据解决方案的残余和梯度提出适应性抽样方法(ASMs),我们首先根据解决方案的剩余和梯度提出仅剩的基于ASM算法。在这种方法中,我们首先通过使用少量剩余点来训练神经网络,并将计算域分为一定的子领域,然后在子领域增加具有最大平均绝对值的子领域残留物(PINNs)中的新残留点。为了更好地捕捉到解决方案的绝对值,我们提议根据解决方案的剩余和梯度来进一步开发第二种类型的ASM算法(ASM II), 仅通过SM IM II 的原始和直方程式来有效测量解决方案的精度。