A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable n...A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.展开更多
This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-d...This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries(VC-MKdV)equation.For the forward problem of the VC-MKdV equation,the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed.Furthermore,the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation.As for the data-driven inverse problem of the VC-MKdV equation,the author introduces a parallel neural networks to separately train the solution function and coefficient function,successfully addressing the function discovery problem of the VC-MKdV equation.To further enhance the network’s generalization ability and noise robustness,the author incorporates two regularization strategies into the PPINNs.An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation,and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.展开更多
The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechani...The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechanism,which can guarantee the global regularity of smooth solutions to the radiative Euler equations for small initial data.Thus a problem of interest is to see to what extent does the viscosity and/or thermal conductivity influence the global regularity of smooth solutions to the one-dimensional radiative Euler equations for large initial data.For results in this direction,it is shown in[30]that,for a class of state equations,even if a special class of thermal conductivity is further added to the radiative Euler equations,its smooth solutions will still blow up in finite time for large initial data.The main purpose of this paper focuses on the case when both viscosity and thermal conductivity are considered.We first show that,for the state equations and the heat conductivity considered in[30],if the viscosity is further taken into account,the corresponding radiative Navier-Stokes equations does admit a unique global smooth solution for any large initial data provided that the viscosity is a smooth function of the density satisfying certain growth conditions as the density tends to zero and infinity.Moreover,we also show that similar result still holds for the case when the thermodynamics variables satisfy the state equations for ideal polytropic gases,the heat conductivity takes the form studied in[30],and the viscosity is assumed to satisfy the same conditions imposed in the first result.展开更多
The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ...The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ≥ 2 and A （u） = -div（a （x, u, u）） is a Leray-Lions operator defined from W 0 1,p（x） （Ω） in to its dual W-1,p＇（x） （Ω）. However the second part concerns the existence solution, of the following setting nonlinear elliptic problems A（u）＋g（x,u, u） = u in Ω, u = 0 on Ω. We will give some regularity results for these solutions.展开更多
针对协方差分析描述函数法(Covariance Analysis Describing Function Technique,CADET)在分析存在内部参数摄动的不确定系统时精度不高的问题,提出了一种分析存在内部参数摄动的导弹姿态控制系统新型精度分析方法。结合传统的CADET方法...针对协方差分析描述函数法(Covariance Analysis Describing Function Technique,CADET)在分析存在内部参数摄动的不确定系统时精度不高的问题,提出了一种分析存在内部参数摄动的导弹姿态控制系统新型精度分析方法。结合传统的CADET方法,对含有参数摄动的广义非线性项进行统计线性化,得到状态均值和协方差的增广传播方程,采用改进的CADET方法对某型号导弹的姿态控制系统进行了数学仿真。仿真结果表明了改进的CADET方法可快速、有效分析存在外部干扰和内部参数摄动系统的精度。展开更多
In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coe...In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L∞and the initial velocity is small enough in B3,10(R3). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L1(R3)and B3,10(R3) respectively.展开更多
文摘A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.
文摘This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries(VC-MKdV)equation.For the forward problem of the VC-MKdV equation,the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed.Furthermore,the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation.As for the data-driven inverse problem of the VC-MKdV equation,the author introduces a parallel neural networks to separately train the solution function and coefficient function,successfully addressing the function discovery problem of the VC-MKdV equation.To further enhance the network’s generalization ability and noise robustness,the author incorporates two regularization strategies into the PPINNs.An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation,and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.
基金supported by the NSFC(12001495)supported by the NSFC(12221001,12371225)the Science and Technology Department of Hubei Province(2020DFH002)。
文摘The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechanism,which can guarantee the global regularity of smooth solutions to the radiative Euler equations for small initial data.Thus a problem of interest is to see to what extent does the viscosity and/or thermal conductivity influence the global regularity of smooth solutions to the one-dimensional radiative Euler equations for large initial data.For results in this direction,it is shown in[30]that,for a class of state equations,even if a special class of thermal conductivity is further added to the radiative Euler equations,its smooth solutions will still blow up in finite time for large initial data.The main purpose of this paper focuses on the case when both viscosity and thermal conductivity are considered.We first show that,for the state equations and the heat conductivity considered in[30],if the viscosity is further taken into account,the corresponding radiative Navier-Stokes equations does admit a unique global smooth solution for any large initial data provided that the viscosity is a smooth function of the density satisfying certain growth conditions as the density tends to zero and infinity.Moreover,we also show that similar result still holds for the case when the thermodynamics variables satisfy the state equations for ideal polytropic gases,the heat conductivity takes the form studied in[30],and the viscosity is assumed to satisfy the same conditions imposed in the first result.
文摘The first part of this paper is devoted to study the existence of solution for nonlinear p（x） elliptic problem A（u） =u in Ω, u = 0 on Ω, with a right-hand side measure, where Ω is a bounded open set of RN, N ≥ 2 and A （u） = -div（a （x, u, u）） is a Leray-Lions operator defined from W 0 1,p（x） （Ω） in to its dual W-1,p＇（x） （Ω）. However the second part concerns the existence solution, of the following setting nonlinear elliptic problems A（u）＋g（x,u, u） = u in Ω, u = 0 on Ω. We will give some regularity results for these solutions.
文摘针对协方差分析描述函数法(Covariance Analysis Describing Function Technique,CADET)在分析存在内部参数摄动的不确定系统时精度不高的问题,提出了一种分析存在内部参数摄动的导弹姿态控制系统新型精度分析方法。结合传统的CADET方法,对含有参数摄动的广义非线性项进行统计线性化,得到状态均值和协方差的增广传播方程,采用改进的CADET方法对某型号导弹的姿态控制系统进行了数学仿真。仿真结果表明了改进的CADET方法可快速、有效分析存在外部干扰和内部参数摄动系统的精度。
基金supported by the National Natural Science Foundation of China(Nos.11731007,11688101)Innovation Grant from National Center for Mathematics and Interdisciplinary Sciences
文摘In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L∞and the initial velocity is small enough in B3,10(R3). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L1(R3)and B3,10(R3) respectively.
