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We consider solving the forward and inverse PDEs which have sharp solutionsusing physics-informed neural networks (PINNs) in this work. In particular, tobetter capture the sharpness of the solution, we propose adaptive samplingmethods (ASMs) based on the residual and the gradient of the solution. We firstpresent 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 pointsand divide the computational domain into a certain number of sub-domains, wethen add new residual points in the sub-domain which has the largest meanabsolute value of the residual, and those points which have largest absolutevalues 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 onboth the residual and the gradient of the solution due to the fact that onlythe residual may be not able to efficiently capture the sharpness of thesolution. The procedure of ASM II is almost the same as that of ASM I exceptthat in ASM II, we add new residual points which not only have large residualbut also large gradient. To demonstrate the effectiveness of the presentmethods, we employ both ASM I and ASM II to solve a number of PDEs, includingBurger equation, compressible Euler equation, Poisson equation over an L-shapedomain as well as high-dimensional Poisson equation. It has been shown from thenumerical results that the sharp solutions can be well approximated by usingeither ASM I or ASM II algorithm, and both methods deliver much more accuratesolution than original PINNs with the same number of residual points. Moreover,the ASM II algorithm has better performance in terms of accuracy, efficiencyand stability compared with the ASM I algorithm.

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions