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

作　　者：Zhiping MAO Xuhui MENG 

机构地区：[1]School of Mathematical Sciences,Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing,Xiamen University,Xiamen 361005,Fujian Province,China [2]Institute of Interdisciplinary Research for Mathematics and Applied Science,School of Mathematics and Statistics,Huazhong University of Science and Technology,Wuhan 430074,China

出　　处：《Applied Mathematics and Mechanics(English Edition)》2023年第7期1069-1084,共16页应用数学和力学（英文版）

基　　金：Project supported by the National Key R&D Program of China(No.2022YFA1004504);the National Natural Science Foundation of China(Nos.12171404 and 12201229);the Fundamental Research Funds for Central Universities of China(No.20720210037)。

摘　　要：We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundar

关 键 词：physics-informed neural network(PINN) adaptive sampling high-dimension L-shape Poisson equation accuracy 

分 类 号：O343[理学—固体力学]