Based on the newly-developed element energy projection （EEP） method for computation of super-convergent results in one-dimensional finite element method （FEM）, the task of self-adaptive FEM analysis was converted ...Based on the newly-developed element energy projection （EEP） method for computation of super-convergent results in one-dimensional finite element method （FEM）, the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.展开更多
The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value...The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.展开更多
We propose and numerically demonstrate an ultrafast real-time ordinary differential equation （ODE） computing unit in optical field based on a silicon microring resonator, operating in the critical coupling region as...We propose and numerically demonstrate an ultrafast real-time ordinary differential equation （ODE） computing unit in optical field based on a silicon microring resonator, operating in the critical coupling region as an optical temporal differentiator. As basic building blocks of a signal processing system, a subtractor and a splitter are included in the proposed structure. This scheme is featured with high speed, compact size and integration on a siliconon-insulator （SOl） wafer. The size of this computing unit is only 35 μm × 45 μm. In this paper, the performance of the proposed structure is theoretically studied and analyzed by numerical simulations.展开更多
It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of ...It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of thickening-system data make this possible.However,the unique properties of thickening systems,such as the non-linearities,long-time delays,partially observed data,and continuous time evolution pose challenges on building data-driven predictive models.To address the above challenges,we establish an integrated,deep-learning,continuous time network structure that consists of a sequential encoder,a state decoder,and a derivative module to learn the deterministic state space model from thickening systems.Using a case study,we examine our methods with a tailing thickener manufactured by the FLSmidth installed with massive sensors and obtain extensive experimental results.The results demonstrate that the proposed continuous-time model with the sequential encoder achieves better prediction performances than the existing discrete-time models and reduces the negative effects from long time delays by extracting features from historical system trajectories.The proposed method also demonstrates outstanding performances for both short and long term prediction tasks with the two proposed derivative types.展开更多
This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution...This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.展开更多
The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However...The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations(ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM.展开更多
This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by d...This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by decomposition approach.Here,the proposed method is an hybrid of the perturbation theory and decomposition method.In this approach,the approximate solution is slihtly perturbed with the MNPs to ensure absolute convergence.Nonlinear cases are first treated by decomposition.The method is,easy to execute with well-posed mathematical formulae.The existence and convergence of the method is also presented explicitly.Resulting numerical evidences show that the proposed method,in comparison with the Adomian Decomposition Method(ADM),Homotpy Pertubation Method and the exact solution is reliable,efficient and accuarate.展开更多
The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear probl...The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.展开更多
A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differe...A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).展开更多
基金Project supported by the National Natural Science Foundation of China (No.50278046)
文摘Based on the newly-developed element energy projection （EEP） method for computation of super-convergent results in one-dimensional finite element method （FEM）, the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.
基金Supported by the Scientific Research Foundation for Advisor Program of Higher Education of Gansu Province(1009-6)Supported by the Scientific Research Foundation for Youth Scholars of Hexi University(qn201015)
文摘The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.
基金Acknowledgements This work was partly supported by the National Natural Science Foundation of China （Grant Nos. 61077052 and 61125504）, Foundation of Ministry of Education of China （No. 20110073110012）, and Science and Technology Commission of Shanghai Municipality （No. 11530700400）.
文摘We propose and numerically demonstrate an ultrafast real-time ordinary differential equation （ODE） computing unit in optical field based on a silicon microring resonator, operating in the critical coupling region as an optical temporal differentiator. As basic building blocks of a signal processing system, a subtractor and a splitter are included in the proposed structure. This scheme is featured with high speed, compact size and integration on a siliconon-insulator （SOl） wafer. The size of this computing unit is only 35 μm × 45 μm. In this paper, the performance of the proposed structure is theoretically studied and analyzed by numerical simulations.
基金supported by National Key Research and Development Program of China(2019YFC0605300)the National Natural Science Foundation of China(61873299,61902022,61972028)+2 种基金Scientific and Technological Innovation Foundation of Shunde Graduate School,University of Science and Technology Beijing(BK21BF002)Macao Science and Technology Development Fund under Macao Funding Scheme for Key R&D Projects(0025/2019/AKP)Macao Science and Technology Development Fund(0015/2020/AMJ)。
文摘It is crucial to predict the outputs of a thickening system,including the underflow concentration(UC)and mud pressure,for optimal control of the process.The proliferation of industrial sensors and the availability of thickening-system data make this possible.However,the unique properties of thickening systems,such as the non-linearities,long-time delays,partially observed data,and continuous time evolution pose challenges on building data-driven predictive models.To address the above challenges,we establish an integrated,deep-learning,continuous time network structure that consists of a sequential encoder,a state decoder,and a derivative module to learn the deterministic state space model from thickening systems.Using a case study,we examine our methods with a tailing thickener manufactured by the FLSmidth installed with massive sensors and obtain extensive experimental results.The results demonstrate that the proposed continuous-time model with the sequential encoder achieves better prediction performances than the existing discrete-time models and reduces the negative effects from long time delays by extracting features from historical system trajectories.The proposed method also demonstrates outstanding performances for both short and long term prediction tasks with the two proposed derivative types.
文摘This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.
基金Project supported by the National Natural Science Foundation of China (Nos. 12132010 and12021002)the Natural Science Foundation of Tianjin of China (No. 19JCZDJC38800)。
文摘The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations(ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM.
文摘This paper focuses on the application of Mamadu-Njoseh polynomials(MNPs)as basis functions for the solution of singular initial value problems in the second-order ordinary differential equations in a perturbation by decomposition approach.Here,the proposed method is an hybrid of the perturbation theory and decomposition method.In this approach,the approximate solution is slihtly perturbed with the MNPs to ensure absolute convergence.Nonlinear cases are first treated by decomposition.The method is,easy to execute with well-posed mathematical formulae.The existence and convergence of the method is also presented explicitly.Resulting numerical evidences show that the proposed method,in comparison with the Adomian Decomposition Method(ADM),Homotpy Pertubation Method and the exact solution is reliable,efficient and accuarate.
基金supported by the National Natural Science Foundation of China(Nos.51378293,51078199,50678093,and 50278046)the Program for Changjiang Scholars and the Innovative Research Team in University of China(No.IRT00736)
文摘The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.
基金Project supported by the National Natural Science Foundation of China(No.11521091)
文摘A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).
