Runge-Kutta型多...解非定常偏微分方程（英文）_陈泽斌.pdf

第40卷第2期2023年3月新疆大学学报（自然科学版）（中英文）Journal of Xinjiang University(Natural Science Edition in Chinese and English)Vol.40,No.2Mar.,2023Multi-Scale Neural Networks Based onRunge-Kutta Method for SolvingUnsteady Partial Differential EquationsCHEN Zebin,FENG Xinlong(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)Abstract:This paper proposes the multi-scale neural networks method based on Runge-Kutta to solve unsteady partial dif-ferential equations.The method uses q-order Runge-Kutta to construct the time iteratione scheme,and further establishes thetotal loss function of multiple time steps,which is to realize the parameter sharing of neural networks with multiple time steps,and to predict the function value at any moment in the time domain.Besides,the m-scaling factor is adopted to speed up theconvergence of the loss function and improve the accuracy of the numerical solution.Finally,several numerical experiments arepresented to demonstrate the effectiveness of the proposed method.Key words:unsteady partial differential equations;q-order Runge-Kutta method;multi-scale neural networks;m-scaling factor;high accuracyDOI：10.13568/ki.651094.651316.2022.06.25.0001CLC number：O175Document Code：AArticle ID：2096-7675(2023)02-0142-08引文格式：陈泽斌,冯新龙.Runge-Kutta 型多尺度神经网络求解非定常偏微分方程J.新疆大学学报(自然科学版)(中英文),2023,40(2):142-149.英文引文格式：CHEN Zebin,FENG Xinlong.Multi-scale neural networks based on Runge-Kutta method for solving unsteadypartial differential equationsJ.Journal of Xinjiang University(Natural Science Edition in Chinese and English),2023,40(2):142-149.Runge-Kutta型多尺度神经网络求解非定常偏微分方程陈泽斌,冯新龙(新疆大学 数学与系统科学学院，新疆 乌鲁木齐830017)摘要：提出了基于 Runge-Kutta 的多尺度神经网络方法求解非定常偏微分方程.利用 q 阶 Runge-Kutta 构造时间迭代格式,通过建立多时间步的总损失函数,实现多时间步的神经网络参数共享,并预测时域内任意时刻的函数值.同时采用m-缩放因子加快损失函数收敛,提高数值解精度.最后,给出了若干数值实验验证所提方法的有效性关键词：非定常偏微分方程;q 阶 Runge-Kutta 法;多尺度神经网络;m-缩放因子;高精度0IntroductionDeep learning has achieved satisfactory results in searching technology,natural language processing,image processing,recommendation system,personalization technology,etc.In recent years,deep learning has been successfully applied tosolve partial differential equations and has been deeply promoted.Compared with the finite element method1and thefinite difference method2,deep learning as a meshless method,can mitigate the curse of dimensionality when solving high-dimensional partial differential equations.So it is more convenient to establish a solution framework for high-dimensionalpartial differential equations.E etc3proposed the Deep-Ritz method based on deep neural networks for numerically solvingvariational problems,which was insensitive to the dimension of the problem and could be used to solve high-dimensional pr-Received Date：2022-06-25Foundation Item：This work was supported by Open Project of Key Laboratory of Xinjiang“Machine learning for incompressible magnetohydrodynam-ics models”(2020D04002).Biography：CHEN Zebin(1995-),male,master student,research fields:deep learning to solve partial differential equations.Corresponding author：FENG Xinlong(1976-),male,professor,research field:numerical solutions of partial differential equations,E-mail:fxl-.No.2CHEN Zebin,et al:Multi-Scale Neural Networks Based on Runge-Kutta Method for Solving Unsteady Partial Differential Equations143oblems.Physics-Informed Neural Networks(PINNs)4used the automatic differentiation technique5for the first time toembed the residual of the equation into the loss function of the neural networks,and obtained the numerical solution of theequation by minimizing the loss function.PINNs is a new numerical method for solving partial differential equations,whichmakes full use of the physical information contained in the PDEs.PINNs have attracted the attention of many scholars,andliterature67shows theoretical convergence of PINNs for certain classes of PDEs.Multi-scale DNN8proposed the idea ofradial scaling in the frequency domain,which had the ability to approximate high-frequency and high-dimensional functions,and could accelerate the convergence speed of the loss function.Recently,the research on deep learning algorithms for nonlinear unsteady partial differential equations have attracted theattention of many scholars.The time-discrete model of PINNs can still guarantee the stability and high precision of numericalsolutions when using large time steps,but it needs high computational costs.DeLISA added the physical information ofthe governing equations into the time iteration scheme,and introduced time-dependent input to achieve continuous-timeprediction without a large number of interior points9.On this basis,this paper proposes a multi-scale neural networksalgorithm integrating Runge-Kutta method.On the one hand,the algorithm constructs a time iteration scheme to build thetotal loss function of multiple time steps,thus realizes the sharing of neural networks parameters in multiple time steps tosave computational costs.It can also predict the function value at any time in the time