The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN app...The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN approaches generally utilize a fully connected network(FCN)architecture that is susceptible to overfitting,training instability,and gradient vanishing as the network depth increases.These challenges result in accuracy bottlenecks in the solution.In response to these issues,the residual-based resample physics-informed neural network(R2-PINN)is proposed.It is an improved PINN architecture that replaces the FCN with a convolutional neural network with a shortcut(S-CNN).It incorporates skip connections to facilitate gradient propagation between network layers.Additionally,the incorporation of the residual adaptive resampling(RAR)mechanism dynamically increases the number of sampling points.This,in turn,enhances the spatial representation capabilities and overall predictive accuracy of the model.The experimental results illustrate that our approach significantly improves the convergence capability of the model and achieves high-precision predictions of the physical fields.Compared with conventional FCN-based PINN methods,R 2-PINN effectively overcomes the limitations inherent in current methods.Thus,it provides more accurate and robust solutions for neutron diffusion equations.展开更多
Physics-informed neural networks(PINNs)are vital for machine learning and exhibit significant advantages when handling complex physical problems.The PINN method can rapidly predict ^(220)Rn progeny concentration and i...Physics-informed neural networks(PINNs)are vital for machine learning and exhibit significant advantages when handling complex physical problems.The PINN method can rapidly predict ^(220)Rn progeny concentration and is very important for regulating and measuring this property.To construct a PINN model,training data are typically preprocessed;however,this approach changes the physical characteristics of the data,with the preprocessed data potentially no longer directly conforming to the original physical equations.As a result,the original physical equations cannot be directly employed in the PINN.Consequently,an effective method for transforming physical equations is crucial for accurately constraining PINNs to model the ^(220)Rn progeny concentration prediction.This study presents an equation adaptation approach for neural networks,which is designed to improve prediction of ^(220)Rn progeny concentration.Five neural network models based on three architectures are established:a classical network,a physics-informed network without equation adaptation,and a physics-informed network with equation adaptation.The transport equation of the ^(220)Rn progeny concentration is transformed via equation adaption and integrated with the PINN model.The compatibility and robustness of the model with equation adaption is then analyzed.The results show that PINNs with equation adaption converge consistently with classical neural networks in terms of the training and validation loss and achieve the same level of prediction accuracy.This outcome indicates that the proposed method can be integrated into the neural network architecture.Moreover,the prediction performance of classical neural networks declines significantly when interference data are encountered,whereas the PINNs with equation adaption exhibit stable prediction accuracy.This performance demonstrates that the proposed method successfully harnesses the constraining power of physical equations,significantly enhancing the robustness of the resultant PINN models.Thus,the use of a physics-informed network with equation adaption can guarantee accurate prediction of ^(220)Rn progeny concentration.展开更多
Physics-informed neural networks(PINNs),as a novel artificial intelligence method for solving partial differential equations,are applicable to solve both forward and inverse problems.This study evaluates the performan...Physics-informed neural networks(PINNs),as a novel artificial intelligence method for solving partial differential equations,are applicable to solve both forward and inverse problems.This study evaluates the performance of PINNs in solving the temperature diffusion equation of the seawater across six scenarios,including forward and inverse problems under three different boundary conditions.Results demonstrate that PINNs achieved consistently higher accuracy with the Dirichlet and Neumann boundary conditions compared to the Robin boundary condition for both forward and inverse problems.Inaccurate weighting of terms in the loss function can reduce model accuracy.Additionally,the sensitivity of model performance to the positioning of sampling points varied between different boundary conditions.In particular,the model under the Dirichlet boundary condition exhibited superior robustness to variations in point positions during the solutions of inverse problems.In contrast,for the Neumann and Robin boundary conditions,accuracy declines when points were sampled from identical positions or at the same time.Subsequently,the Argo observations were used to reconstruct the vertical diffusion of seawater temperature in the north-central Pacific for the applicability of PINNs in the real ocean.The PINNs successfully captured the vertical diffusion characteristics of seawater temperature,reflected the seasonal changes of vertical temperature under different topographic conditions,and revealed the influence of topography on the temperature diffusion coefficient.The PINNs were proved effective in solving the temperature diffusion equation of seawater with limited data,providing a promising technique for simulating or predicting ocean phenomena using sparse observations.展开更多
Owing to intensified globalization and informatization,the structures of the urban scale hierarchy and urban networks between cities have become increasingly intertwined,resulting in different spatial effects.Therefor...Owing to intensified globalization and informatization,the structures of the urban scale hierarchy and urban networks between cities have become increasingly intertwined,resulting in different spatial effects.Therefore,this paper analyzes the spatial interaction between urban scale hierarchy and urban networks in China from 2019 to 2023,drawing on Baidu migration data and employing a spatial simultaneous equation model.The results reveal a significant positive spatial correlation between cities with higher hierarchy and those with greater network centrality.Within a static framework,we identify a positive interaction between urban scale hierarchy and urban network centrality,while their spatial cross-effects manifest as negative neighborhood interactions based on geographical distance and positive cross-scale interactions shaped by network connections.Within a dynamic framework,changes in urban scale hierarchy and urban networks are mutually reinforcing,thereby widening disparities within the urban hierarchy.Furthermore,an increase in a city’s network centrality had a dampening effect on the population growth of neighboring cities and network-connected cities.This study enhances understanding of the spatial organisation of urban systems and offers insights for coordinated regional development.展开更多
Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay di...Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.展开更多
Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-i...Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients.The numerical simulations are carried out with different initial and boundary conditions,in which the relative L2-norm errors are all around 10^(−4).The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations.The dynamic behavior of solitons and their interaction are studied in the colliding and chasing processes for the Boussinesq equation.More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.展开更多
In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symm...In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symmetric points,numerical simulations are conducted to one-soliton and two-soliton solutions of a discrete KdV equation,as well as a one-soliton solution of a discrete Toda lattice equation.Compared with the existing discrete deep learning approach,the numerical results reveal that within the specified spatiotemporal domain,the prediction accuracy by SDE-PINN is excellent regardless of the interior or extrapolation prediction,with a significant reduction in training time.The proposed data enhancement technique and symmetric structure development provides a new perspective for the deep learning approach to solve discrete NLEs.The newly proposed SDE-PINN can also be applied to solve continuous nonlinear equations and other discrete NLEs numerically.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
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 ...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 boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.展开更多
From the analytical dynamics point of view, this paper develops an optimal control framework to synchronize networked multibody systems using the fundamental equation of mechanics. A novel robust control law derived f...From the analytical dynamics point of view, this paper develops an optimal control framework to synchronize networked multibody systems using the fundamental equation of mechanics. A novel robust control law derived from the framework is then used to achieve complete synchronization of networked identical or non-identical multibody systems formulated with Lagrangian dynamics. A distinctive feature of the developed control strategy is the introduction of network structures into the control requirement. The control law consists of two components, the first describing the architecture of the network and the second denoting an active feedback control strategy. A corresponding stability analysis is performed by the algebraic graph theory. A representative network composed of ten identical or non-identical gyroscopes is used as an illustrative example. Numerical simulation of the systems with three kinds of network structures, including global coupling, nearest-neighbour, and small-world networks, is given to demonstrate effectiveness of the proposed control methodology.展开更多
Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonl...Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well.Second,the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper,we propose a new architecture that combines the deep residual neural network with some underlying physical laws.Using the sine-Gordon equation as an example,we show that the numerical result is in good agreement with the exact soliton solution.In addition,a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.展开更多
In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a ste...In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.展开更多
Novel distributed parameter neural networks are proposed for solving partial differential equations, and their dynamic performances are studied in Hilbert space. The locally connected neural networks are obtained by s...Novel distributed parameter neural networks are proposed for solving partial differential equations, and their dynamic performances are studied in Hilbert space. The locally connected neural networks are obtained by separating distributed parameter neural networks. Two simulations are also given. Both theoretical and computed results illustrate that the distributed parameter neural networks are effective and efficient for solving partial differential equation problems.展开更多
This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In par...This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.展开更多
The closed form of solutions of Kac-van Moerbeke lattice and self-dual network equations are considered by proposing transformations based on Riccati equation, using symbolic computation. In contrast to the numerical ...The closed form of solutions of Kac-van Moerbeke lattice and self-dual network equations are considered by proposing transformations based on Riccati equation, using symbolic computation. In contrast to the numerical computation of travelling wave solutions for differential difference equations, our method obtains exact solutions which have physical relevance.展开更多
The N-fold Darboux transformation(DT) T_n^([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1,...The N-fold Darboux transformation(DT) T_n^([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t^2).展开更多
Bayesian networks （BN） have many advantages over other methods in ecological modeling, and have become an increasingly popular modeling tool. However, BN are flawed in regard to building models based on inadequate e...Bayesian networks （BN） have many advantages over other methods in ecological modeling, and have become an increasingly popular modeling tool. However, BN are flawed in regard to building models based on inadequate existing knowledge. To overcome this limitation, we propose a new method that links BN with structural equation modeling （SEM）. In this method, SEM is used to improve the model structure for BN. This method was used to simulate coastal phytoplankton dynamics in the Bohai Bay. We demonstrate that this hybrid approach minimizes the need for expert elicitation, generates more reasonable structures for BN models, and increases the BN model＇s accuracy and reliability. These results suggest that the inclusion of SEM for testing and verifying the theoretical structure during the initial construction stage improves the effectiveness of BN models, especially for complex eco-environment systems. The results also demonstrate that in the Bohai Bay, while phytoplankton biomass has the greatest influence on phytoplankton dynamics, the impact of nutrients on phytoplankton dynamics is larger than the influence of the physical environment in summer. Furthermore, although the Redfield ratio indicates that phosphorus should be the primary nutrient limiting factor, our results show that silicate plays the most important role in regulating phytoplankton dynamics in the Bohai Bay.展开更多
Growth rate of Yarrowia lipolytica NCIM 3589 was observed in a fermentation medium consisting of peptone, yeast extract, sodium chloride. Logistic equation was fitted to the growth data (time vs. biomass concentration...Growth rate of Yarrowia lipolytica NCIM 3589 was observed in a fermentation medium consisting of peptone, yeast extract, sodium chloride. Logistic equation was fitted to the growth data (time vs. biomass concentration) and compared with the prediction given by Artificial Neural Networks (ANN). ANN was found to be superior in describing growth characteristics. A single MATLAB programme is developed to fit the growth data by logistic equation and ANN.展开更多
We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. ...We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.展开更多
This paper focuses on a very important point which consists in evaluating experimental data prior to their use for chemical process designs. Hexafluoropropylene P, ρ, T data measured at 11 temperatures from 263 to 36...This paper focuses on a very important point which consists in evaluating experimental data prior to their use for chemical process designs. Hexafluoropropylene P, ρ, T data measured at 11 temperatures from 263 to 362 K and at pressures up to 10 MPa have been examined through a consistency test presented herein and based on the use of a methodology implying both neural networks and Virial equation. Such a methodology appears as very powerful to identify erroneous data and could be conveniently handled for quick checks of databases previously to modeling through classical thermodynamic models and equations of state. As an application to liquid and vapor phase densities of hexafluoropropylene, a more reliable database is provided after removing out layer data.展开更多
基金supported by the Science and Technology on Reactor System Design Technology Laboratory(No.LRSDT12023108)supported in part by the Chongqing Postdoctoral Science Foundation(No.cstc2021jcyj-bsh0252)+2 种基金the National Natural Science Foundation of China(No.12005030)Sichuan Province to unveil the list of marshal industry common technology research projects(No.23jBGOV0001)Special Program for Stabilizing Support to Basic Research of National Basic Research Institutes(No.WDZC-2023-05-03-05).
文摘The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN approaches generally utilize a fully connected network(FCN)architecture that is susceptible to overfitting,training instability,and gradient vanishing as the network depth increases.These challenges result in accuracy bottlenecks in the solution.In response to these issues,the residual-based resample physics-informed neural network(R2-PINN)is proposed.It is an improved PINN architecture that replaces the FCN with a convolutional neural network with a shortcut(S-CNN).It incorporates skip connections to facilitate gradient propagation between network layers.Additionally,the incorporation of the residual adaptive resampling(RAR)mechanism dynamically increases the number of sampling points.This,in turn,enhances the spatial representation capabilities and overall predictive accuracy of the model.The experimental results illustrate that our approach significantly improves the convergence capability of the model and achieves high-precision predictions of the physical fields.Compared with conventional FCN-based PINN methods,R 2-PINN effectively overcomes the limitations inherent in current methods.Thus,it provides more accurate and robust solutions for neutron diffusion equations.
基金supported by the National Natural Science Foundation of China(Nos.12375310,118750356,and 62006110)Graduate Research and Innovation Projects of Hunan Province(CX20230964).
文摘Physics-informed neural networks(PINNs)are vital for machine learning and exhibit significant advantages when handling complex physical problems.The PINN method can rapidly predict ^(220)Rn progeny concentration and is very important for regulating and measuring this property.To construct a PINN model,training data are typically preprocessed;however,this approach changes the physical characteristics of the data,with the preprocessed data potentially no longer directly conforming to the original physical equations.As a result,the original physical equations cannot be directly employed in the PINN.Consequently,an effective method for transforming physical equations is crucial for accurately constraining PINNs to model the ^(220)Rn progeny concentration prediction.This study presents an equation adaptation approach for neural networks,which is designed to improve prediction of ^(220)Rn progeny concentration.Five neural network models based on three architectures are established:a classical network,a physics-informed network without equation adaptation,and a physics-informed network with equation adaptation.The transport equation of the ^(220)Rn progeny concentration is transformed via equation adaption and integrated with the PINN model.The compatibility and robustness of the model with equation adaption is then analyzed.The results show that PINNs with equation adaption converge consistently with classical neural networks in terms of the training and validation loss and achieve the same level of prediction accuracy.This outcome indicates that the proposed method can be integrated into the neural network architecture.Moreover,the prediction performance of classical neural networks declines significantly when interference data are encountered,whereas the PINNs with equation adaption exhibit stable prediction accuracy.This performance demonstrates that the proposed method successfully harnesses the constraining power of physical equations,significantly enhancing the robustness of the resultant PINN models.Thus,the use of a physics-informed network with equation adaption can guarantee accurate prediction of ^(220)Rn progeny concentration.
基金Supported by the National Key Research and Development Program of China(No.2023YFC3008200)the Independent Research Project of Southern Marine Science and Engineering Guangdong Laboratory(Zhuhai)(No.SML2022SP505)。
文摘Physics-informed neural networks(PINNs),as a novel artificial intelligence method for solving partial differential equations,are applicable to solve both forward and inverse problems.This study evaluates the performance of PINNs in solving the temperature diffusion equation of the seawater across six scenarios,including forward and inverse problems under three different boundary conditions.Results demonstrate that PINNs achieved consistently higher accuracy with the Dirichlet and Neumann boundary conditions compared to the Robin boundary condition for both forward and inverse problems.Inaccurate weighting of terms in the loss function can reduce model accuracy.Additionally,the sensitivity of model performance to the positioning of sampling points varied between different boundary conditions.In particular,the model under the Dirichlet boundary condition exhibited superior robustness to variations in point positions during the solutions of inverse problems.In contrast,for the Neumann and Robin boundary conditions,accuracy declines when points were sampled from identical positions or at the same time.Subsequently,the Argo observations were used to reconstruct the vertical diffusion of seawater temperature in the north-central Pacific for the applicability of PINNs in the real ocean.The PINNs successfully captured the vertical diffusion characteristics of seawater temperature,reflected the seasonal changes of vertical temperature under different topographic conditions,and revealed the influence of topography on the temperature diffusion coefficient.The PINNs were proved effective in solving the temperature diffusion equation of seawater with limited data,providing a promising technique for simulating or predicting ocean phenomena using sparse observations.
基金Under the auspices of the National Natural Science Foundation of China(No.42371222,41971167)Fundamental Scientific Research Funds of Central China Normal University(No.CCNU24ZZ120)。
文摘Owing to intensified globalization and informatization,the structures of the urban scale hierarchy and urban networks between cities have become increasingly intertwined,resulting in different spatial effects.Therefore,this paper analyzes the spatial interaction between urban scale hierarchy and urban networks in China from 2019 to 2023,drawing on Baidu migration data and employing a spatial simultaneous equation model.The results reveal a significant positive spatial correlation between cities with higher hierarchy and those with greater network centrality.Within a static framework,we identify a positive interaction between urban scale hierarchy and urban network centrality,while their spatial cross-effects manifest as negative neighborhood interactions based on geographical distance and positive cross-scale interactions shaped by network connections.Within a dynamic framework,changes in urban scale hierarchy and urban networks are mutually reinforcing,thereby widening disparities within the urban hierarchy.Furthermore,an increase in a city’s network centrality had a dampening effect on the population growth of neighboring cities and network-connected cities.This study enhances understanding of the spatial organisation of urban systems and offers insights for coordinated regional development.
文摘Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.
基金supported by the National Natural Science Foundation of China under Grant No.12475204.
文摘Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients.The numerical simulations are carried out with different initial and boundary conditions,in which the relative L2-norm errors are all around 10^(−4).The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations.The dynamic behavior of solitons and their interaction are studied in the colliding and chasing processes for the Boussinesq equation.More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.
基金supported by the National Natural Science Foundation of China(Grant No.12071042)the Beijing Natural Science Foundation(Grant No.1202004)Promoting the Development of University Classification-Student Innovation and Entrepreneurship Training Programme(Grant No.5112410857)。
文摘In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symmetric points,numerical simulations are conducted to one-soliton and two-soliton solutions of a discrete KdV equation,as well as a one-soliton solution of a discrete Toda lattice equation.Compared with the existing discrete deep learning approach,the numerical results reveal that within the specified spatiotemporal domain,the prediction accuracy by SDE-PINN is excellent regardless of the interior or extrapolation prediction,with a significant reduction in training time.The proposed data enhancement technique and symmetric structure development provides a new perspective for the deep learning approach to solve discrete NLEs.The newly proposed SDE-PINN can also be applied to solve continuous nonlinear equations and other discrete NLEs numerically.
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).
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)。
文摘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 boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.
基金Project supported by the National Natural Science Foundation of China(Nos.10972129 and 11272191)the Specialized Research Foundation for the Doctoral Program of Higher Education(No.200802800015)+1 种基金the Science and Technology Project of High Schools of Shandong Province(No.J15LJ07)the Shandong Provincial Natural Science Foundation(No.ZR2015FL026)
文摘From the analytical dynamics point of view, this paper develops an optimal control framework to synchronize networked multibody systems using the fundamental equation of mechanics. A novel robust control law derived from the framework is then used to achieve complete synchronization of networked identical or non-identical multibody systems formulated with Lagrangian dynamics. A distinctive feature of the developed control strategy is the introduction of network structures into the control requirement. The control law consists of two components, the first describing the architecture of the network and the second denoting an active feedback control strategy. A corresponding stability analysis is performed by the algebraic graph theory. A representative network composed of ten identical or non-identical gyroscopes is used as an illustrative example. Numerical simulation of the systems with three kinds of network structures, including global coupling, nearest-neighbour, and small-world networks, is given to demonstrate effectiveness of the proposed control methodology.
基金The authors gratefully acknowledge the support of the National Natural Science Foundation of China(Grant No.11675054)Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)Science and Technology Commission of Shanghai Municipality(Grant No.18dz2271000).
文摘Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well.Second,the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper,we propose a new architecture that combines the deep residual neural network with some underlying physical laws.Using the sine-Gordon equation as an example,we show that the numerical result is in good agreement with the exact soliton solution.In addition,a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.
文摘In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.
文摘Novel distributed parameter neural networks are proposed for solving partial differential equations, and their dynamic performances are studied in Hilbert space. The locally connected neural networks are obtained by separating distributed parameter neural networks. Two simulations are also given. Both theoretical and computed results illustrate that the distributed parameter neural networks are effective and efficient for solving partial differential equation problems.
文摘This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.
基金The project supported by ＂973＂ Project under Grant No.2004CB318000, the Doctor Start-up Foundation of Liaoning Province of China under Grant No. 20041066, and the Science Research Plan of Liaoning Education Bureau under Grant No. 2004F099
文摘The closed form of solutions of Kac-van Moerbeke lattice and self-dual network equations are considered by proposing transformations based on Riccati equation, using symbolic computation. In contrast to the numerical computation of travelling wave solutions for differential difference equations, our method obtains exact solutions which have physical relevance.
基金Supported by the Natural Science Foundation of Zhejiang Province under Grant No.LY15A010005the Natural Science Foundation of Ningbo under Grant No.2018A610197+1 种基金the NSF of China under Grant No.11671219K.C.Wong Magna Fund in Ningbo University
文摘The N-fold Darboux transformation(DT) T_n^([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t^2).
基金supported by the Natural Science Foundation of Tianjin(Grant No.16JCYBJC23000)the Open Foundation of the Key Laboratory for Ecological Environment in Coastal Areas of the State Oceanic Administration(Grant No.201604)Science and Technology Foundation for Young Scholars from Tianjin Fisheries Bureau(Grant No.J2014-05)
文摘Bayesian networks （BN） have many advantages over other methods in ecological modeling, and have become an increasingly popular modeling tool. However, BN are flawed in regard to building models based on inadequate existing knowledge. To overcome this limitation, we propose a new method that links BN with structural equation modeling （SEM）. In this method, SEM is used to improve the model structure for BN. This method was used to simulate coastal phytoplankton dynamics in the Bohai Bay. We demonstrate that this hybrid approach minimizes the need for expert elicitation, generates more reasonable structures for BN models, and increases the BN model＇s accuracy and reliability. These results suggest that the inclusion of SEM for testing and verifying the theoretical structure during the initial construction stage improves the effectiveness of BN models, especially for complex eco-environment systems. The results also demonstrate that in the Bohai Bay, while phytoplankton biomass has the greatest influence on phytoplankton dynamics, the impact of nutrients on phytoplankton dynamics is larger than the influence of the physical environment in summer. Furthermore, although the Redfield ratio indicates that phosphorus should be the primary nutrient limiting factor, our results show that silicate plays the most important role in regulating phytoplankton dynamics in the Bohai Bay.
文摘Growth rate of Yarrowia lipolytica NCIM 3589 was observed in a fermentation medium consisting of peptone, yeast extract, sodium chloride. Logistic equation was fitted to the growth data (time vs. biomass concentration) and compared with the prediction given by Artificial Neural Networks (ANN). ANN was found to be superior in describing growth characteristics. A single MATLAB programme is developed to fit the growth data by logistic equation and ANN.
文摘We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.
文摘This paper focuses on a very important point which consists in evaluating experimental data prior to their use for chemical process designs. Hexafluoropropylene P, ρ, T data measured at 11 temperatures from 263 to 362 K and at pressures up to 10 MPa have been examined through a consistency test presented herein and based on the use of a methodology implying both neural networks and Virial equation. Such a methodology appears as very powerful to identify erroneous data and could be conveniently handled for quick checks of databases previously to modeling through classical thermodynamic models and equations of state. As an application to liquid and vapor phase densities of hexafluoropropylene, a more reliable database is provided after removing out layer data.
