Physics-informed neural networks(PINNs)have emerged as a promising class of scientific machine learning techniques that integrate governing physical laws into neural network training.Their ability to enforce different...Physics-informed neural networks(PINNs)have emerged as a promising class of scientific machine learning techniques that integrate governing physical laws into neural network training.Their ability to enforce differential equations,constitutive relations,and boundary conditions within the loss function provides a physically grounded alternative to traditional data-driven models,particularly for solid and structural mechanics,where data are often limited or noisy.This review offers a comprehensive assessment of recent developments in PINNs,combining bibliometric analysis,theoretical foundations,application-oriented insights,and methodological innovations.A biblio-metric survey indicates a rapid increase in publications on PINNs since 2018,with prominent research clusters focused on numerical methods,structural analysis,and forecasting.Building upon this trend,the review consolidates advance-ments across five principal application domains,including forward structural analysis,inverse modeling and parameter identification,structural and topology optimization,assessment of structural integrity,and manufacturing processes.These applications are propelled by substantial methodological advancements,encompassing rigorous enforcement of boundary conditions,modified loss functions,adaptive training,domain decomposition strategies,multi-fidelity and transfer learning approaches,as well as hybrid finite element–PINN integration.These advances address recurring challenges in solid mechanics,such as high-order governing equations,material heterogeneity,complex geometries,localized phenomena,and limited experimental data.Despite remaining challenges in computational cost,scalability,and experimental validation,PINNs are increasingly evolving into specialized,physics-aware tools for practical solid and structural mechanics applications.展开更多
Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicate...Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicated problems such as irregular boundary conditions(BCs)and discontinuous or high-frequency behaviors remain persistent challenges for PINNs.For these reasons,we propose a novel two-phase framework,where a neural network is first trained to represent shape functions that can capture the irregularity of BCs in the first phase,and then these neural network-based shape functions are used to construct boundary shape functions(BSFs)that exactly satisfy both essential and natural BCs in PINNs in the second phase.This scheme is integrated into both the strong-form and energy PINN approaches,thereby improving the quality of solution prediction in the cases of irregular BCs.In addition,this study examines the benefits and limitations of these approaches in handling discontinuous and high-frequency problems.Overall,our method offers a unified and flexible solution framework that addresses key limitations of existing PINN methods with higher accuracy and stability for general PDE problems in solid mechanics.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
The Internet of Things(IoT)ecosystem faces growing security challenges because it is projected to have 76.88 billion devices by 2025 and $1.4 trillion market value by 2027,operating in distributed networks with resour...The Internet of Things(IoT)ecosystem faces growing security challenges because it is projected to have 76.88 billion devices by 2025 and $1.4 trillion market value by 2027,operating in distributed networks with resource limitations and diverse system architectures.The current conventional intrusion detection systems(IDS)face scalability problems and trust-related issues,but blockchain-based solutions face limitations because of their low transaction throughput(Bitcoin:7 TPS(Transactions Per Second),Ethereum:15-30 TPS)and high latency.The research introduces MBID(Multi-Tier Blockchain Intrusion Detection)as a groundbreaking Multi-Tier Blockchain Intrusion Detection System with AI-Enhanced Detection,which solves the problems in huge IoT networks.The MBID system uses a four-tier architecture that includes device,edge,fog,and cloud layers with blockchain implementations and Physics-Informed Neural Networks(PINNs)for edge-based anomaly detection and a dual consensus mechanism that uses Honesty-based Distributed Proof-of-Authority(HDPoA)and Delegated Proof of Stake(DPoS).The system achieves scalability and efficiency through the combination of dynamic sharding and Interplanetary File System(IPFS)integration.Experimental evaluations demonstrate exceptional performance,achieving a detection accuracy of 99.84%,an ultra-low false positive rate of 0.01% with a False Negative Rate of 0.15%,and a near-instantaneous edge detection latency of 0.40 ms.The system demonstrated an aggregate throughput of 214.57 TPS in a 3-shard configuration,providing a clear,evidence-based path for horizontally scaling to support overmillions of devices with exceeding throughput.The proposed architecture represents a significant advancement in blockchain-based security for IoT networks,effectively balancing the trade-offs between scalability,security,and decentralization.展开更多
Physics-informed neural networks(PINNs)have shown remarkable prospects in solving the forward and inverse problems involving partial differential equations(PDEs).The method embeds PDEs into the neural network by calcu...Physics-informed neural networks(PINNs)have shown remarkable prospects in solving the forward and inverse problems involving partial differential equations(PDEs).The method embeds PDEs into the neural network by calculating the PDE loss at a set of collocation points,providing advantages such as meshfree and more convenient adaptive sampling.However,when solving PDEs using nonuniform collocation points,PINNs still face challenge regarding inefficient convergence of PDE residuals or even failure.In this work,we first analyze the ill-conditioning of the PDE loss in PINNs under nonuniform collocation points.To address the issue,we define volume weighting residual and propose volume weighting physics-informed neural networks(VW-PINNs).Through weighting the PDE residuals by the volume that the collocation points occupy within the computational domain,we embed explicitly the distribution characteristics of collocation points in the loss evaluation.The fast and sufficient convergence of the PDE residuals for the problems involving nonuniform collocation points is guaranteed.Considering the meshfree characteristics of VW-PINNs,we also develop a volume approximation algorithm based on kernel density estimation to calculate the volume of the collocation points.We validate the universality of VW-PINNs by solving the forward problems involving flow over a circular cylinder and flow over the NACA0012 airfoil under different inflow conditions,where conventional PINNs fail.By solving the Burgers’equation,we verify that VW-PINNs can enhance the efficiency of existing the adaptive sampling method in solving the forward problem by three times,and can reduce the relative L 2 error of conventional PINNs in solving the inverse problem by more than one order of magnitude.展开更多
For the diagnostics and health management of lithium-ion batteries,numerous models have been developed to understand their degradation characteristics.These models typically fall into two categories:data-driven models...For the diagnostics and health management of lithium-ion batteries,numerous models have been developed to understand their degradation characteristics.These models typically fall into two categories:data-driven models and physical models,each offering unique advantages but also facing limitations.Physics-informed neural networks(PINNs)provide a robust framework to integrate data-driven models with physical principles,ensuring consistency with underlying physics while enabling generalization across diverse operational conditions.This study introduces a PINN-based approach to reconstruct open circuit voltage(OCV)curves and estimate key ageing parameters at both the cell and electrode levels.These parameters include available capacity,electrode capacities,and lithium inventory capacity.The proposed method integrates OCV reconstruction models as functional components into convolutional neural networks(CNNs)and is validated using a public dataset.The results reveal that the estimated ageing parameters closely align with those obtained through offline OCV tests,with errors in reconstructed OCV curves remaining within 15 mV.This demonstrates the ability of the method to deliver fast and accurate degradation diagnostics at the electrode level,advancing the potential for precise and efficient battery health management.展开更多
Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static...Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static problems;however,the simultaneous enforcement of I/BCs in dynamic problems remains challenging.To overcome this limitation,a novel approach called decoupled physics-informed neural network(d PINN)is proposed in this work.The d PINN operates based on the core idea of converting a partial differential equation(PDE)to a system of ordinary differential equations(ODEs)via the space-time decoupled formulation.To this end,the latent solution is expressed in the form of a linear combination of approximation functions and coefficients,where approximation functions are admissible and coefficients are unknowns of time that must be solved.Subsequently,the system of ODEs is obtained by implementing the weighted-residual form of the original PDE over the spatial domain.A multi-network structure is used to parameterize the set of coefficient functions,and the loss function of d PINN is established based on minimizing the residuals of the gained ODEs.In this scheme,the decoupled formulation leads to the independent handling of I/BCs.Accordingly,the BCs are automatically satisfied based on suitable selections of admissible functions.Meanwhile,the original ICs are replaced by the Galerkin form of the ICs concerning unknown coefficients,and the neural network(NN)outputs are modified to satisfy the gained ICs.Several benchmark problems involving different types of PDEs and I/BCs are used to demonstrate the superior performance of d PINN compared with regular PINN in terms of solution accuracy and computational cost.展开更多
This research introduces a spectrum-based physics-informed neural network(SP-PINN)model to significantly improve the accuracy of calculation of two-phase flow parameters,surpassing existing methods that have limitatio...This research introduces a spectrum-based physics-informed neural network(SP-PINN)model to significantly improve the accuracy of calculation of two-phase flow parameters,surpassing existing methods that have limitations in global and continuous data sampling.SP-PINNs address the challenges of traditional methods in terms of continuous sampling by integrating the spectral analysis and pressure correction into the Navier-Stokes(N-S)equations,enhancing the predictive accuracy especially in critical regions like gas-phase boundaries and velocity peaks.The novel introduction of a pressure-correction module within SP-PINNs mitigates prediction errors,achieving a substantial reduction to 1‰compared with the conventional physics-informed neural network(PINN)approaches.Experimental applications validate the model’s ability to accurately and rapidly predict flow parameters with different sampling time intervals,with the computation time of predicting unsampled data less than 0.01 s.Such advancements signify a 100-fold improvement over traditional DNS calculations,underscoring the model’s potential in the real-time calculation and analysis of multiphase flow dynamics.展开更多
Physics-informed neural networks(PINNs)have shown considerable promise for performing numerical simulations in fluid mechanics.They provide mesh-free,end-to-end approaches by embedding physical laws into their loss fu...Physics-informed neural networks(PINNs)have shown considerable promise for performing numerical simulations in fluid mechanics.They provide mesh-free,end-to-end approaches by embedding physical laws into their loss functions.However,when addressing complex flow problems,PINNs still face some challenges such as activation saturation and vanishing gradients in deep network training,leading to slow convergence and insufficient prediction accuracy.We present physics-informed neural networks incorporating lattice Boltzmann method optimized by tanh robust weight initialization(T-PINN-LBM)to address these challenges.This approach fuses the mesoscopic lattice Boltzmann model with the automatic differentiation framework of PINNs.It also implements a tanh robust weight initialization method derived from fixed point analysis.This model effectively mitigates activation and gradient decay in deep networks,improving convergence speed and data efficiency in multiscale flow simulations.We validate the effectiveness of the model on the classical arithmetic example of lid-driven cavity flow.Compared to the traditional Xavier initialized PINN and PINN-LBM,T-PINNLBM reduces the mean absolute error(MAE)by one order of magnitude at the same network depth and maintains stable convergence in deeper networks.The results demonstrate that this model can accurately capture complex flow structures without prior data,providing a new feasible pathway for data-free driven fluid simulation.展开更多
This paper proposes a physics-informed neural network(PINN)framework to analyze the nonlinear buckling behavior of a three-dimensional(3D)FG porous,slender beam resting on a Winkler-Pasternak foundation.PINNs need muc...This paper proposes a physics-informed neural network(PINN)framework to analyze the nonlinear buckling behavior of a three-dimensional(3D)FG porous,slender beam resting on a Winkler-Pasternak foundation.PINNs need much less training data to obtain high accuracy using a straightforward network.The powerful tool used in this work can handle any class of PDEs.We use the deep learning platform TensorFlow and DeepXDE library to design our network.In this study,the PINNs framework takes information from the governing differential equations of the beam system and the data from boundary conditions and outputs the critical nonlinear buckling load.The mathematical model is developed using Hamilton’s principle,considering geometry’s nonlinearity.The accuracy of the modeling framework is carefully examined by applying it to various boundary condition cases as well as the physical parameters such as 3D FG indexes on the nonlinear mechanical behaviors.Finally,the PINNs results are validated with those extracted from the generalized differential quadrature method(GDQM).It is found that the proposed PINN framework can characterize the nonlinear buckling behavior of 3D FG porous,slender beams with satisfactory accuracy.Furthermore,PINN is presented to accurately predict the nonlinear buckling behavior of the beam up to 71 times faster than the numerical method.展开更多
Multi-scale system remains a classical scientific problem in fluid dynamics,biology,etc.In the present study,a scheme of multi-scale Physics-informed neural networks is proposed to solve the boundary layer flow at hig...Multi-scale system remains a classical scientific problem in fluid dynamics,biology,etc.In the present study,a scheme of multi-scale Physics-informed neural networks is proposed to solve the boundary layer flow at high Reynolds numbers without any data.The flow is divided into several regions with different scales based on Prandtl's boundary theory.Different regions are solved with governing equations in different scales.The method of matched asymptotic expansions is used to make the flow field continuously.A flow on a semi infinite flat plate at a high Reynolds number is considered a multi-scale problem because the boundary layer scale is much smaller than the outer flow scale.The results are compared with the reference numerical solutions,which show that the msPINNs can solve the multi-scale problem of the boundary layer in high Reynolds number flows.This scheme can be developed for more multi-scale problems in the future.展开更多
With the explosive growth of computational resources and data generation,deep machine learning has been successfully employed in various applications.One important and emerging scientific application of deep learning ...With the explosive growth of computational resources and data generation,deep machine learning has been successfully employed in various applications.One important and emerging scientific application of deep learning involves solving differential equations.Here,physics-informed neural networks(PINNs)are developed to solve the differential equations associated with a specific scientific problem.As such,algorithms for solving the differential equations by embedding their initial and boundary conditions in the cost function of the artificial neural networks using algorithmic differentiation must also be developed.In this study,various PINNs are adopted to estimate the stresses in the tablets and the interphase of a single lap joint.The proposed model is represented by two fourth-order non-homogeneous coupled partial differential equations,with the axial stresses in the upper and lower tablets adopted as the dependent variables.The axial stresses are a function of the tablet length,which presents the independent variable.Therefore,the axial stresses in the tablets are estimated by solving the coupled partial differential equations when subjected to the boundary conditions,whereas the remaining stress components are expressed in terms of axial stresses.The results obtained using the developed methodology are validated using the results obtained via MAPLE software.展开更多
Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the los...Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the loss function.The performance of PINNs is generally affected by both training and sampling.Specifically,training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs,and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished.However,a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category,namely,time-dependent PDEs,where temporal information plays a key role in the algorithms used.There is one method,called Causal PINN,that considers temporal causality at the training level but not special temporal utilization at the sampling level.Incorporating temporal knowledge into sampling remains to be studied.To fill this gap,we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality.By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain,we provide a practical solution by incorporating temporal information into sampling.Numerical experiments of several nonlinear time-dependent PDEs,including the Cahn–Hilliard,Korteweg–de Vries,Allen–Cahn and wave equations,show that our proposed sampling method can improve the performance.We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods,especially when points are limited.展开更多
Partial differential equations(PDEs)are important tools for scientific research and are widely used in various fields.However,it is usually very difficult to obtain accurate analytical solutions of PDEs,and numerical ...Partial differential equations(PDEs)are important tools for scientific research and are widely used in various fields.However,it is usually very difficult to obtain accurate analytical solutions of PDEs,and numerical methods to solve PDEs are often computationally intensive and very time-consuming.In recent years,Physics Informed Neural Networks(PINNs)have been successfully applied to find numerical solutions of PDEs and have shown great potential.All the while,solitary waves have been of great interest to researchers in the field of nonlinear science.In this paper,we perform numerical simulations of solitary wave solutions of several PDEs using improved PINNs.The improved PINNs not only incorporate constraints on the control equations to ensure the interpretability of the prediction results,which is important for physical field simulations,in addition,an adaptive activation function is introduced.By introducing hyperparameters in the activation function to change the slope of the activation function to avoid the disappearance of the gradient,computing time is saved thereby speeding up training.In this paper,the m Kd V equation,the improved Boussinesq equation,the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the p-g BKP equation are selected for study,and the errors of the simulation results are analyzed to assess the accuracy of the predicted solitary wave solution.The experimental results show that the improved PINNs are significantly better than the traditional PINNs with shorter training time but more accurate prediction results.The improved PINNs improve the training speed by more than 1.5 times compared with the traditional PINNs,while maintaining the prediction error less than 10~(-2)in this order of magnitude.展开更多
Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep lea...Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations.Among them,physics-informed neural networks(PINNs)are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena.In the field of nonlinear science,solitary waves and rogue waves have been important research topics.In this paper,we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints.In addition,we employ meta-learning optimization to speed up the training process.We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves.We evaluate the accuracy of the prediction results by error analysis.The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.展开更多
We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatio...We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh.Specifically,we optimize the neural network by minimizing the loss function to satisfy the differential operators,initial condition and boundary condition.Then,we prove the convergence of the loss function and the convergence of the neural network.In addition,the feasibility and effectiveness of the method are verified by the results of numerical experiments,and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.展开更多
Fusion and fission are two important phenomena that have been experimentally observed in many real physical models.In this paper,we investigate the two phenomena in the(2+1)-dimensional Hirota-Satsuma-Ito equation via...Fusion and fission are two important phenomena that have been experimentally observed in many real physical models.In this paper,we investigate the two phenomena in the(2+1)-dimensional Hirota-Satsuma-Ito equation via the physics-informed neural networks(PINN)method.By choosing suitable physically constrained initial boundary conditions,the data-driven fusion and fission solutions are obtained for the first time.Dynamical behaviors and error analysis of these solutions are investigated via illustratively numerical figures,which show that good results are achieved.It is pointed out that the PINN method adopted here can be effectively used to construct the data-driven fusion and fission solutions for other nonlinear integrable equations.Based on the powerful predictive capability of the PINN method and wide applications of fusion and fission in many physical areas,it is hoped that the data-driven solutions obtained here will be helpful for experts to predict or explain related physical phenomena.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
Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions.However,material identification is a challenging task,especially when the ch...Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions.However,material identification is a challenging task,especially when the characteristic of the material is highly nonlinear in nature,as is common in biological tissue.In this work,we identify unknown material properties in continuum solid mechanics via physics-informed neural networks(PINNs).To improve the accuracy and efficiency of PINNs,we develop efficient strategies to nonuniformly sample observational data.We also investigate different approaches to enforce Dirichlet-type boundary conditions(BCs)as soft or hard constraints.Finally,we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space.The estimated material parameters achieve relative errors of less than 1%.As such,this work is relevant to diverse applications,including optimizing structural integrity and developing novel materials.展开更多
A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp...A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.展开更多
基金funded by National Research Council of Thailand(contract No.N42A671047).
文摘Physics-informed neural networks(PINNs)have emerged as a promising class of scientific machine learning techniques that integrate governing physical laws into neural network training.Their ability to enforce differential equations,constitutive relations,and boundary conditions within the loss function provides a physically grounded alternative to traditional data-driven models,particularly for solid and structural mechanics,where data are often limited or noisy.This review offers a comprehensive assessment of recent developments in PINNs,combining bibliometric analysis,theoretical foundations,application-oriented insights,and methodological innovations.A biblio-metric survey indicates a rapid increase in publications on PINNs since 2018,with prominent research clusters focused on numerical methods,structural analysis,and forecasting.Building upon this trend,the review consolidates advance-ments across five principal application domains,including forward structural analysis,inverse modeling and parameter identification,structural and topology optimization,assessment of structural integrity,and manufacturing processes.These applications are propelled by substantial methodological advancements,encompassing rigorous enforcement of boundary conditions,modified loss functions,adaptive training,domain decomposition strategies,multi-fidelity and transfer learning approaches,as well as hybrid finite element–PINN integration.These advances address recurring challenges in solid mechanics,such as high-order governing equations,material heterogeneity,complex geometries,localized phenomena,and limited experimental data.Despite remaining challenges in computational cost,scalability,and experimental validation,PINNs are increasingly evolving into specialized,physics-aware tools for practical solid and structural mechanics applications.
基金Project supported by the Basic Science Research Program through the National Research Foundation(NRF)of Korea funded by the Ministry of Science and ICT(No.RS-2024-00337001)。
文摘Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicated problems such as irregular boundary conditions(BCs)and discontinuous or high-frequency behaviors remain persistent challenges for PINNs.For these reasons,we propose a novel two-phase framework,where a neural network is first trained to represent shape functions that can capture the irregularity of BCs in the first phase,and then these neural network-based shape functions are used to construct boundary shape functions(BSFs)that exactly satisfy both essential and natural BCs in PINNs in the second phase.This scheme is integrated into both the strong-form and energy PINN approaches,thereby improving the quality of solution prediction in the cases of irregular BCs.In addition,this study examines the benefits and limitations of these approaches in handling discontinuous and high-frequency problems.Overall,our method offers a unified and flexible solution framework that addresses key limitations of existing PINN methods with higher accuracy and stability for general PDE problems in solid mechanics.
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supported in part by Multimedia University under the Research Fellow Grant MMUI/250008in part by Telekom Research&Development Sdn Bhd underGrantRDTC/241149Princess Nourah bint Abdulrahman University Researchers Supporting Project number(PNURSP2025R140),Princess Nourah bint Abdulrahman University,Riyadh,Saudi Arabia.
文摘The Internet of Things(IoT)ecosystem faces growing security challenges because it is projected to have 76.88 billion devices by 2025 and $1.4 trillion market value by 2027,operating in distributed networks with resource limitations and diverse system architectures.The current conventional intrusion detection systems(IDS)face scalability problems and trust-related issues,but blockchain-based solutions face limitations because of their low transaction throughput(Bitcoin:7 TPS(Transactions Per Second),Ethereum:15-30 TPS)and high latency.The research introduces MBID(Multi-Tier Blockchain Intrusion Detection)as a groundbreaking Multi-Tier Blockchain Intrusion Detection System with AI-Enhanced Detection,which solves the problems in huge IoT networks.The MBID system uses a four-tier architecture that includes device,edge,fog,and cloud layers with blockchain implementations and Physics-Informed Neural Networks(PINNs)for edge-based anomaly detection and a dual consensus mechanism that uses Honesty-based Distributed Proof-of-Authority(HDPoA)and Delegated Proof of Stake(DPoS).The system achieves scalability and efficiency through the combination of dynamic sharding and Interplanetary File System(IPFS)integration.Experimental evaluations demonstrate exceptional performance,achieving a detection accuracy of 99.84%,an ultra-low false positive rate of 0.01% with a False Negative Rate of 0.15%,and a near-instantaneous edge detection latency of 0.40 ms.The system demonstrated an aggregate throughput of 214.57 TPS in a 3-shard configuration,providing a clear,evidence-based path for horizontally scaling to support overmillions of devices with exceeding throughput.The proposed architecture represents a significant advancement in blockchain-based security for IoT networks,effectively balancing the trade-offs between scalability,security,and decentralization.
基金supported by the National Natural Science Foundation of China(Grant No.92152301)the National Key Research and Development Program of China(Grant No.2022YFB4300200)the Shaanxi Provincial Key Research and Development Program(Grant No.2023-ZDLGY-27).
文摘Physics-informed neural networks(PINNs)have shown remarkable prospects in solving the forward and inverse problems involving partial differential equations(PDEs).The method embeds PDEs into the neural network by calculating the PDE loss at a set of collocation points,providing advantages such as meshfree and more convenient adaptive sampling.However,when solving PDEs using nonuniform collocation points,PINNs still face challenge regarding inefficient convergence of PDE residuals or even failure.In this work,we first analyze the ill-conditioning of the PDE loss in PINNs under nonuniform collocation points.To address the issue,we define volume weighting residual and propose volume weighting physics-informed neural networks(VW-PINNs).Through weighting the PDE residuals by the volume that the collocation points occupy within the computational domain,we embed explicitly the distribution characteristics of collocation points in the loss evaluation.The fast and sufficient convergence of the PDE residuals for the problems involving nonuniform collocation points is guaranteed.Considering the meshfree characteristics of VW-PINNs,we also develop a volume approximation algorithm based on kernel density estimation to calculate the volume of the collocation points.We validate the universality of VW-PINNs by solving the forward problems involving flow over a circular cylinder and flow over the NACA0012 airfoil under different inflow conditions,where conventional PINNs fail.By solving the Burgers’equation,we verify that VW-PINNs can enhance the efficiency of existing the adaptive sampling method in solving the forward problem by three times,and can reduce the relative L 2 error of conventional PINNs in solving the inverse problem by more than one order of magnitude.
基金supported by the Beijing Natural Science Foundation(Grant No.L223013)。
文摘For the diagnostics and health management of lithium-ion batteries,numerous models have been developed to understand their degradation characteristics.These models typically fall into two categories:data-driven models and physical models,each offering unique advantages but also facing limitations.Physics-informed neural networks(PINNs)provide a robust framework to integrate data-driven models with physical principles,ensuring consistency with underlying physics while enabling generalization across diverse operational conditions.This study introduces a PINN-based approach to reconstruct open circuit voltage(OCV)curves and estimate key ageing parameters at both the cell and electrode levels.These parameters include available capacity,electrode capacities,and lithium inventory capacity.The proposed method integrates OCV reconstruction models as functional components into convolutional neural networks(CNNs)and is validated using a public dataset.The results reveal that the estimated ageing parameters closely align with those obtained through offline OCV tests,with errors in reconstructed OCV curves remaining within 15 mV.This demonstrates the ability of the method to deliver fast and accurate degradation diagnostics at the electrode level,advancing the potential for precise and efficient battery health management.
基金Project supported by the Basic Science Research Program through the National Research Foundation(NRF)of Korea funded by the Ministry of Science and ICT(No.RS-2024-00337001)。
文摘Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static problems;however,the simultaneous enforcement of I/BCs in dynamic problems remains challenging.To overcome this limitation,a novel approach called decoupled physics-informed neural network(d PINN)is proposed in this work.The d PINN operates based on the core idea of converting a partial differential equation(PDE)to a system of ordinary differential equations(ODEs)via the space-time decoupled formulation.To this end,the latent solution is expressed in the form of a linear combination of approximation functions and coefficients,where approximation functions are admissible and coefficients are unknowns of time that must be solved.Subsequently,the system of ODEs is obtained by implementing the weighted-residual form of the original PDE over the spatial domain.A multi-network structure is used to parameterize the set of coefficient functions,and the loss function of d PINN is established based on minimizing the residuals of the gained ODEs.In this scheme,the decoupled formulation leads to the independent handling of I/BCs.Accordingly,the BCs are automatically satisfied based on suitable selections of admissible functions.Meanwhile,the original ICs are replaced by the Galerkin form of the ICs concerning unknown coefficients,and the neural network(NN)outputs are modified to satisfy the gained ICs.Several benchmark problems involving different types of PDEs and I/BCs are used to demonstrate the superior performance of d PINN compared with regular PINN in terms of solution accuracy and computational cost.
基金Supported by the National Natural Science Foundation of China(No.62304022)。
文摘This research introduces a spectrum-based physics-informed neural network(SP-PINN)model to significantly improve the accuracy of calculation of two-phase flow parameters,surpassing existing methods that have limitations in global and continuous data sampling.SP-PINNs address the challenges of traditional methods in terms of continuous sampling by integrating the spectral analysis and pressure correction into the Navier-Stokes(N-S)equations,enhancing the predictive accuracy especially in critical regions like gas-phase boundaries and velocity peaks.The novel introduction of a pressure-correction module within SP-PINNs mitigates prediction errors,achieving a substantial reduction to 1‰compared with the conventional physics-informed neural network(PINN)approaches.Experimental applications validate the model’s ability to accurately and rapidly predict flow parameters with different sampling time intervals,with the computation time of predicting unsampled data less than 0.01 s.Such advancements signify a 100-fold improvement over traditional DNS calculations,underscoring the model’s potential in the real-time calculation and analysis of multiphase flow dynamics.
基金supported by the National Natural Science Foundation of China(Grant Nos.12474453,12174085,and 12404530).
文摘Physics-informed neural networks(PINNs)have shown considerable promise for performing numerical simulations in fluid mechanics.They provide mesh-free,end-to-end approaches by embedding physical laws into their loss functions.However,when addressing complex flow problems,PINNs still face some challenges such as activation saturation and vanishing gradients in deep network training,leading to slow convergence and insufficient prediction accuracy.We present physics-informed neural networks incorporating lattice Boltzmann method optimized by tanh robust weight initialization(T-PINN-LBM)to address these challenges.This approach fuses the mesoscopic lattice Boltzmann model with the automatic differentiation framework of PINNs.It also implements a tanh robust weight initialization method derived from fixed point analysis.This model effectively mitigates activation and gradient decay in deep networks,improving convergence speed and data efficiency in multiscale flow simulations.We validate the effectiveness of the model on the classical arithmetic example of lid-driven cavity flow.Compared to the traditional Xavier initialized PINN and PINN-LBM,T-PINNLBM reduces the mean absolute error(MAE)by one order of magnitude at the same network depth and maintains stable convergence in deeper networks.The results demonstrate that this model can accurately capture complex flow structures without prior data,providing a new feasible pathway for data-free driven fluid simulation.
文摘This paper proposes a physics-informed neural network(PINN)framework to analyze the nonlinear buckling behavior of a three-dimensional(3D)FG porous,slender beam resting on a Winkler-Pasternak foundation.PINNs need much less training data to obtain high accuracy using a straightforward network.The powerful tool used in this work can handle any class of PDEs.We use the deep learning platform TensorFlow and DeepXDE library to design our network.In this study,the PINNs framework takes information from the governing differential equations of the beam system and the data from boundary conditions and outputs the critical nonlinear buckling load.The mathematical model is developed using Hamilton’s principle,considering geometry’s nonlinearity.The accuracy of the modeling framework is carefully examined by applying it to various boundary condition cases as well as the physical parameters such as 3D FG indexes on the nonlinear mechanical behaviors.Finally,the PINNs results are validated with those extracted from the generalized differential quadrature method(GDQM).It is found that the proposed PINN framework can characterize the nonlinear buckling behavior of 3D FG porous,slender beams with satisfactory accuracy.Furthermore,PINN is presented to accurately predict the nonlinear buckling behavior of the beam up to 71 times faster than the numerical method.
文摘Multi-scale system remains a classical scientific problem in fluid dynamics,biology,etc.In the present study,a scheme of multi-scale Physics-informed neural networks is proposed to solve the boundary layer flow at high Reynolds numbers without any data.The flow is divided into several regions with different scales based on Prandtl's boundary theory.Different regions are solved with governing equations in different scales.The method of matched asymptotic expansions is used to make the flow field continuously.A flow on a semi infinite flat plate at a high Reynolds number is considered a multi-scale problem because the boundary layer scale is much smaller than the outer flow scale.The results are compared with the reference numerical solutions,which show that the msPINNs can solve the multi-scale problem of the boundary layer in high Reynolds number flows.This scheme can be developed for more multi-scale problems in the future.
基金Project supported by the Science and Engineering Research Board(SERB),Department of Science and Technology(DST),India(No.SRG/2019/001581)。
文摘With the explosive growth of computational resources and data generation,deep machine learning has been successfully employed in various applications.One important and emerging scientific application of deep learning involves solving differential equations.Here,physics-informed neural networks(PINNs)are developed to solve the differential equations associated with a specific scientific problem.As such,algorithms for solving the differential equations by embedding their initial and boundary conditions in the cost function of the artificial neural networks using algorithmic differentiation must also be developed.In this study,various PINNs are adopted to estimate the stresses in the tablets and the interphase of a single lap joint.The proposed model is represented by two fourth-order non-homogeneous coupled partial differential equations,with the axial stresses in the upper and lower tablets adopted as the dependent variables.The axial stresses are a function of the tablet length,which presents the independent variable.Therefore,the axial stresses in the tablets are estimated by solving the coupled partial differential equations when subjected to the boundary conditions,whereas the remaining stress components are expressed in terms of axial stresses.The results obtained using the developed methodology are validated using the results obtained via MAPLE software.
基金Project supported by the Key National Natural Science Foundation of China(Grant No.62136005)the National Natural Science Foundation of China(Grant Nos.61922087,61906201,and 62006238)。
文摘Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the loss function.The performance of PINNs is generally affected by both training and sampling.Specifically,training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs,and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished.However,a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category,namely,time-dependent PDEs,where temporal information plays a key role in the algorithms used.There is one method,called Causal PINN,that considers temporal causality at the training level but not special temporal utilization at the sampling level.Incorporating temporal knowledge into sampling remains to be studied.To fill this gap,we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality.By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain,we provide a practical solution by incorporating temporal information into sampling.Numerical experiments of several nonlinear time-dependent PDEs,including the Cahn–Hilliard,Korteweg–de Vries,Allen–Cahn and wave equations,show that our proposed sampling method can improve the performance.We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods,especially when points are limited.
文摘Partial differential equations(PDEs)are important tools for scientific research and are widely used in various fields.However,it is usually very difficult to obtain accurate analytical solutions of PDEs,and numerical methods to solve PDEs are often computationally intensive and very time-consuming.In recent years,Physics Informed Neural Networks(PINNs)have been successfully applied to find numerical solutions of PDEs and have shown great potential.All the while,solitary waves have been of great interest to researchers in the field of nonlinear science.In this paper,we perform numerical simulations of solitary wave solutions of several PDEs using improved PINNs.The improved PINNs not only incorporate constraints on the control equations to ensure the interpretability of the prediction results,which is important for physical field simulations,in addition,an adaptive activation function is introduced.By introducing hyperparameters in the activation function to change the slope of the activation function to avoid the disappearance of the gradient,computing time is saved thereby speeding up training.In this paper,the m Kd V equation,the improved Boussinesq equation,the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the p-g BKP equation are selected for study,and the errors of the simulation results are analyzed to assess the accuracy of the predicted solitary wave solution.The experimental results show that the improved PINNs are significantly better than the traditional PINNs with shorter training time but more accurate prediction results.The improved PINNs improve the training speed by more than 1.5 times compared with the traditional PINNs,while maintaining the prediction error less than 10~(-2)in this order of magnitude.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.42005003 and 41475094).
文摘Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations.Among them,physics-informed neural networks(PINNs)are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena.In the field of nonlinear science,solitary waves and rogue waves have been important research topics.In this paper,we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints.In addition,we employ meta-learning optimization to speed up the training process.We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves.We evaluate the accuracy of the prediction results by error analysis.The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.
基金Project supported in part by the National Natural Science Foundation of China(Grant No.11771259)Shaanxi Provincial Joint Laboratory of Artificial Intelligence(GrantNo.2022JCSYS05)+1 种基金Innovative Team Project of Shaanxi Provincial Department of Education(Grant No.21JP013)Shaanxi Provincial Social Science Fund Annual Project(Grant No.2022D332)。
文摘We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh.Specifically,we optimize the neural network by minimizing the loss function to satisfy the differential operators,initial condition and boundary condition.Then,we prove the convergence of the loss function and the convergence of the neural network.In addition,the feasibility and effectiveness of the method are verified by the results of numerical experiments,and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.
基金supported by the National Natural Science Foundation of China under grant Nos.12371250 and 12205154Jiangsu Provincial Natural Science Foundation under grant Nos.BK20221508 and BK20210380Jiangsu Qinglan High-level Talent Project and High-level Personnel Project under grant No.JSSCBS20210277.
文摘Fusion and fission are two important phenomena that have been experimentally observed in many real physical models.In this paper,we investigate the two phenomena in the(2+1)-dimensional Hirota-Satsuma-Ito equation via the physics-informed neural networks(PINN)method.By choosing suitable physically constrained initial boundary conditions,the data-driven fusion and fission solutions are obtained for the first time.Dynamical behaviors and error analysis of these solutions are investigated via illustratively numerical figures,which show that good results are achieved.It is pointed out that the PINN method adopted here can be effectively used to construct the data-driven fusion and fission solutions for other nonlinear integrable equations.Based on the powerful predictive capability of the PINN method and wide applications of fusion and fission in many physical areas,it is hoped that the data-driven solutions obtained here will be helpful for experts to predict or explain related physical phenomena.
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funded by the Cora Topolewski Cardiac Research Fund at the Children’s Hospital of Philadelphia(CHOP)the Pediatric Valve Center Frontier Program at CHOP+4 种基金the Additional Ventures Single Ventricle Research Fund Expansion Awardthe National Institutes of Health(USA)supported by the program(Nos.NHLBI T32 HL007915 and NIH R01 HL153166)supported by the program(No.NIH R01 HL153166)supported by the U.S.Department of Energy(No.DE-SC0022953)。
文摘Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions.However,material identification is a challenging task,especially when the characteristic of the material is highly nonlinear in nature,as is common in biological tissue.In this work,we identify unknown material properties in continuum solid mechanics via physics-informed neural networks(PINNs).To improve the accuracy and efficiency of PINNs,we develop efficient strategies to nonuniformly sample observational data.We also investigate different approaches to enforce Dirichlet-type boundary conditions(BCs)as soft or hard constraints.Finally,we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space.The estimated material parameters achieve relative errors of less than 1%.As such,this work is relevant to diverse applications,including optimizing structural integrity and developing novel materials.
基金Project supported by the National Natural Science Foundation of China Basic Science Center Program for“Multiscale Problems in Nonlinear Mechanics”(No.11988102)the National Natural Science Foundation of China(No.12202451)。
文摘A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.
