The accurate mechanical analysis of thick-walled pressure vessel structures composed of advanced materials,such as hyperelastic and functionally graded materials(FGMs),is critical for ensuring their safety and optimiz...The accurate mechanical analysis of thick-walled pressure vessel structures composed of advanced materials,such as hyperelastic and functionally graded materials(FGMs),is critical for ensuring their safety and optimizing their design.However,conventional numerical methods can face challenges with the non-linearities inherent in hyperelasticity and the complex spatial variations in FGMs.This paper presents a novel hybrid numerical approach combining Physics-Informed Neural Networks(PINNs)with Finite Element Method(FEM)derived data for the robust analysis of thick-walled,axisymmetric,heterogeneous,hyperelastic pressure vessels with elliptical geometries.A PINN framework incorporating neo-Hookean constitutive relations is developed in MATLAB.To enhance training efficiency and accuracy,the PINN’s loss function is augmented with displacement data obtained from high-fidelity FEM simulations performed in ANSYS.The methodology is rigorously validated by comparing PINN-predicted displacement and von Mises stress fields against ANSYS benchmarks for various scenarios of FGMconfigurations(with material properties varying according to a power law)subjected to internal and external pressurization.The results demonstrate excellent agreement between the proposed hybrid PINN-FEMapproach and conventional FEMsolutions across all test cases,accurately capturing complex deformation patterns and stress concentrations.This study highlights the potential of data-augmented PINNs as an effective and accurate computational tool for tackling complex solid mechanics problems involving non-linearmaterials and significant heterogeneity,offering a promising avenue for future research in engineering design and analysis.展开更多
Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing a...Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing algorithms,mesh-generation is complex,and we cannot tackle high-dimensional problems governed by parametrized NSE.Moreover,solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes.Here,we review flow physics-informed learning,integrating seamlessly data and mathematical models,and implement them using physics-informed neural networks(PINNs).We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows,supersonic flows,and biomedical flows.展开更多
This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampli...This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator,where the truncated Gaussian model has been adopted for estimating the indicator.Here,we present two extensions of that work.The first extension consists in combining with a re-sampling technique,so that the new algorithm can maintain a constant training size.This is achieved through a cosine-annealing,which gradually transforms the sampling of collocation points from uniform to adaptive via the training progress.The second extension is to present the subset simulation(SS)algorithm as the posterior model(instead of the truncated Gaussian model)for estimating the error indicator,which can more effectively estimate the failure probability and generate new effective training points in the failure region.We investigate the performance of the new approach using several challenging problems,and numerical experiments demonstrate a significant improvement over the original algorithm.展开更多
In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field functi...In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.展开更多
We propose a data-driven physics-informed neural networks(PINNs)via task-decomposition(DD-PINNs-TD)for modeling nonlinear thermal-deformation-polarization-carrier(TDPC)coupling mechanical behaviors of piezoelectric se...We propose a data-driven physics-informed neural networks(PINNs)via task-decomposition(DD-PINNs-TD)for modeling nonlinear thermal-deformation-polarization-carrier(TDPC)coupling mechanical behaviors of piezoelectric semiconductors(PSs).By embedding three-dimensional(3D),plate,and beam equations of PS structures into the constraints of the DD-PINNsTD framework,respectively,we develop three representative PINNs that exhibit significant advantages in computational efficiency and accuracy compared to traditional PINNs.Using the proposed DD-PINNs-TD models,we investigate the TDPC coupling responses of PS structures under different loadings.Numerical results demonstrate that the proposed models exhibit accuracy and stability of these models in predicting the nonlinear multi-field coupling mechanical behaviors of PSs.Notably,the plate and beam-theory-based DD-PINNs-TD models achieve superior computational efficiency relative to their 3Dequation-based counterparts.This study establishes a theoretical foundation for analyzing nonlinear multi-field coupling responses in PS stru ctures and has significant practical value in engineering applications.展开更多
Magnetic soft continuum robots(MSCRs)offer transformative potential for minimally invasive procedures due to their high flexibility and magnetic responsiveness.However,reliable and efficient programming of MSCRs for a...Magnetic soft continuum robots(MSCRs)offer transformative potential for minimally invasive procedures due to their high flexibility and magnetic responsiveness.However,reliable and efficient programming of MSCRs for anatomical adaptability and precise tip manipulation remains a key challenge,particularly in navigating tortuous pathways and targeting hard-to-reach lesions.Addressing this,we propose a unified inverse programming framework based on Physics-Informed Neural Networks(PINNs)that simultaneously tackles two critical design objectives in MSCR applications:shape morphing and tip trajectory control.The shape morphing problem involves programming magnetization distributions during fabrication to achieve desired global geometries,while trajectory control is realized by designing time-varying magnetic fields to guide the robot tip along prescribed paths.Leveraging the hard-magnetic elastica model,we reformulate the inverse design challenge into solving a nonlinear ordinary differential equation(ODE).The proposed PINN-based framework seamlessly integrates physical priors into the learning process,enabling rapid convergence while requiring only sparse data.We validate our approach using complex geometries,including shapes resembling the letters“USTC”,and benchmark the results against finite difference(FDM)and finite element method(FEM)simulations.The strong agreement across methods confirms the reliability and accuracy of the PINN-based framework.Our method offers a versatile and computationally efficient tool for the inverse design and control of programmable MSCRs and opens new pathways for data-free,high-fidelity,multi-objective optimization in magnetically actuated soft robotics.展开更多
In this work,we propose a new deep learning approach based on physics-informed neural networks(PINNs),termed parallel parity-time-symmetric PINNs(PPTS-PINNs),to address the inverse problem of determining the PT-symmet...In this work,we propose a new deep learning approach based on physics-informed neural networks(PINNs),termed parallel parity-time-symmetric PINNs(PPTS-PINNs),to address the inverse problem of determining the PT-symmetric potential function in the nonlinear Schrödinger equation(NLSE).By incorporating PT-symmetry constraints and a gradient enhancement strategy,the method effectively improves both the accuracy and stability of solving inverse problems,while preserving the consistency of the physical structure.We conduct systematic numerical experiments on two representative NLSEs under both noise-free and noisy conditions(with noise levels of 1%and 5%)to evaluate the reconstruction performance of the model given fixed observational data.The results demonstrate that PPTS-PINNs can robustly reconstruct the complex-valued potential function across different noise levels,achieving an overall error on the order of 10^(−3).Notably,under high-noise conditions,the combination of PT-symmetry constraints and the gradient enhancement strategy significantly enhances the model’s robustness by mitigating error propagation.Overall,the proposed method exhibits strong adaptability and generalization capabilities in PT-symmetric modeling and noise-resilient learning,offering a novel perspective for solving more complex physical inverse problems.展开更多
We present an efficient physics-informed neural networks(PINNs)framework,termed Adaptive Interface-PINNs(AdaI-PINNs),to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jum...We present an efficient physics-informed neural networks(PINNs)framework,termed Adaptive Interface-PINNs(AdaI-PINNs),to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jumps.This framework is an enhanced version of its predecessor,Interface PINNs or I-PINNs(Sarma et al.[1];https://doi.org/10.1016/j.cma.2024.117135),which involves domain decomposition and assignment of different predefined activation functions to the neural networks in each subdomain across a sharp interface,while keeping all other parameters of the neural networks identical.In AdaI-PINNs,the activation functions vary solely in their slopes,which are trained along with the other parameters of the neural networks.This makes the AdaI-PINNs framework fully automated without requiring preset activation functions.Comparative studies on one-dimensional,two-dimensional,and three-dimensional benchmark elliptic interface problems reveal that AdaI-PINNs outperform I-PINNs,reducing computational costs by 2-6 times while producing similar or better accuracy.展开更多
Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-paramete...Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-parameterized PINNs work remains unknown.This paper presents the first convergence analysis of the overparameterized PINNs for the Laplace equations with Dirichlet boundary conditions.We demonstrate that the convergence rate can be controlled by the weight norm,regardless of the number of parameters in the network.展开更多
文摘The accurate mechanical analysis of thick-walled pressure vessel structures composed of advanced materials,such as hyperelastic and functionally graded materials(FGMs),is critical for ensuring their safety and optimizing their design.However,conventional numerical methods can face challenges with the non-linearities inherent in hyperelasticity and the complex spatial variations in FGMs.This paper presents a novel hybrid numerical approach combining Physics-Informed Neural Networks(PINNs)with Finite Element Method(FEM)derived data for the robust analysis of thick-walled,axisymmetric,heterogeneous,hyperelastic pressure vessels with elliptical geometries.A PINN framework incorporating neo-Hookean constitutive relations is developed in MATLAB.To enhance training efficiency and accuracy,the PINN’s loss function is augmented with displacement data obtained from high-fidelity FEM simulations performed in ANSYS.The methodology is rigorously validated by comparing PINN-predicted displacement and von Mises stress fields against ANSYS benchmarks for various scenarios of FGMconfigurations(with material properties varying according to a power law)subjected to internal and external pressurization.The results demonstrate excellent agreement between the proposed hybrid PINN-FEMapproach and conventional FEMsolutions across all test cases,accurately capturing complex deformation patterns and stress concentrations.This study highlights the potential of data-augmented PINNs as an effective and accurate computational tool for tackling complex solid mechanics problems involving non-linearmaterials and significant heterogeneity,offering a promising avenue for future research in engineering design and analysis.
基金The research of the second author(ZM)was sup-539 ported by the National Natural Science Foundation of China(Grant 54012171404)The last author(GEK)would like to acknowledge support 541 by the Alexander von Humboldt fellowship.
文摘Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing algorithms,mesh-generation is complex,and we cannot tackle high-dimensional problems governed by parametrized NSE.Moreover,solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes.Here,we review flow physics-informed learning,integrating seamlessly data and mathematical models,and implement them using physics-informed neural networks(PINNs).We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows,supersonic flows,and biomedical flows.
基金supported by the NSF of China(No.12171085)This work was supported by the National Key R&D Program of China(2020YFA0712000)+2 种基金the NSF of China(No.12288201)the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA25010404)and the Youth Innovation Promotion Association(CAS).
文摘This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator,where the truncated Gaussian model has been adopted for estimating the indicator.Here,we present two extensions of that work.The first extension consists in combining with a re-sampling technique,so that the new algorithm can maintain a constant training size.This is achieved through a cosine-annealing,which gradually transforms the sampling of collocation points from uniform to adaptive via the training progress.The second extension is to present the subset simulation(SS)algorithm as the posterior model(instead of the truncated Gaussian model)for estimating the error indicator,which can more effectively estimate the failure probability and generate new effective training points in the failure region.We investigate the performance of the new approach using several challenging problems,and numerical experiments demonstrate a significant improvement over the original algorithm.
文摘In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.
基金supported by the National Natural Science Foundation of China(Grant Nos.12172326 and 12192210)the Natural Science Foundation of Zhejiang Province(Grant No.LZ25A020007)the National Key Research and Development Program of China(Grant No.2020YFA0711700)。
文摘We propose a data-driven physics-informed neural networks(PINNs)via task-decomposition(DD-PINNs-TD)for modeling nonlinear thermal-deformation-polarization-carrier(TDPC)coupling mechanical behaviors of piezoelectric semiconductors(PSs).By embedding three-dimensional(3D),plate,and beam equations of PS structures into the constraints of the DD-PINNsTD framework,respectively,we develop three representative PINNs that exhibit significant advantages in computational efficiency and accuracy compared to traditional PINNs.Using the proposed DD-PINNs-TD models,we investigate the TDPC coupling responses of PS structures under different loadings.Numerical results demonstrate that the proposed models exhibit accuracy and stability of these models in predicting the nonlinear multi-field coupling mechanical behaviors of PSs.Notably,the plate and beam-theory-based DD-PINNs-TD models achieve superior computational efficiency relative to their 3Dequation-based counterparts.This study establishes a theoretical foundation for analyzing nonlinear multi-field coupling responses in PS stru ctures and has significant practical value in engineering applications.
基金supported by the National Key Research and Development Program of China(Grant No.2024YFE0215200)the National Natural Science Foundation of China(Grant No.12272369).
文摘Magnetic soft continuum robots(MSCRs)offer transformative potential for minimally invasive procedures due to their high flexibility and magnetic responsiveness.However,reliable and efficient programming of MSCRs for anatomical adaptability and precise tip manipulation remains a key challenge,particularly in navigating tortuous pathways and targeting hard-to-reach lesions.Addressing this,we propose a unified inverse programming framework based on Physics-Informed Neural Networks(PINNs)that simultaneously tackles two critical design objectives in MSCR applications:shape morphing and tip trajectory control.The shape morphing problem involves programming magnetization distributions during fabrication to achieve desired global geometries,while trajectory control is realized by designing time-varying magnetic fields to guide the robot tip along prescribed paths.Leveraging the hard-magnetic elastica model,we reformulate the inverse design challenge into solving a nonlinear ordinary differential equation(ODE).The proposed PINN-based framework seamlessly integrates physical priors into the learning process,enabling rapid convergence while requiring only sparse data.We validate our approach using complex geometries,including shapes resembling the letters“USTC”,and benchmark the results against finite difference(FDM)and finite element method(FEM)simulations.The strong agreement across methods confirms the reliability and accuracy of the PINN-based framework.Our method offers a versatile and computationally efficient tool for the inverse design and control of programmable MSCRs and opens new pathways for data-free,high-fidelity,multi-objective optimization in magnetically actuated soft robotics.
基金supported by National Natural Science Foun-dation of China(Grant No.11501513)Natural Science Foundation of Zhejiang Province(Grant No.LY18A010030).
文摘In this work,we propose a new deep learning approach based on physics-informed neural networks(PINNs),termed parallel parity-time-symmetric PINNs(PPTS-PINNs),to address the inverse problem of determining the PT-symmetric potential function in the nonlinear Schrödinger equation(NLSE).By incorporating PT-symmetry constraints and a gradient enhancement strategy,the method effectively improves both the accuracy and stability of solving inverse problems,while preserving the consistency of the physical structure.We conduct systematic numerical experiments on two representative NLSEs under both noise-free and noisy conditions(with noise levels of 1%and 5%)to evaluate the reconstruction performance of the model given fixed observational data.The results demonstrate that PPTS-PINNs can robustly reconstruct the complex-valued potential function across different noise levels,achieving an overall error on the order of 10^(−3).Notably,under high-noise conditions,the combination of PT-symmetry constraints and the gradient enhancement strategy significantly enhances the model’s robustness by mitigating error propagation.Overall,the proposed method exhibits strong adaptability and generalization capabilities in PT-symmetric modeling and noise-resilient learning,offering a novel perspective for solving more complex physical inverse problems.
基金support from ExxonMobil Corporation to the Subsurface Mechanics and Geo-Energy Laboratory under the grant SP22230020CEEXXU008957The support from the Ministry of Education,Government of India and IIT Madras under the grant SB20210856CEMHRD008957 is also gratefully acknowledged.
文摘We present an efficient physics-informed neural networks(PINNs)framework,termed Adaptive Interface-PINNs(AdaI-PINNs),to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jumps.This framework is an enhanced version of its predecessor,Interface PINNs or I-PINNs(Sarma et al.[1];https://doi.org/10.1016/j.cma.2024.117135),which involves domain decomposition and assignment of different predefined activation functions to the neural networks in each subdomain across a sharp interface,while keeping all other parameters of the neural networks identical.In AdaI-PINNs,the activation functions vary solely in their slopes,which are trained along with the other parameters of the neural networks.This makes the AdaI-PINNs framework fully automated without requiring preset activation functions.Comparative studies on one-dimensional,two-dimensional,and three-dimensional benchmark elliptic interface problems reveal that AdaI-PINNs outperform I-PINNs,reducing computational costs by 2-6 times while producing similar or better accuracy.
基金supported by the National Key Research and Development Program of China(No.2023YFA1000103)the National Natural Science Foundation of China(No.123B2019,No.12125103,No.U24A2002,No.12371424,No.12371441)by the Fundamental Research Funds for the Central Universities.
文摘Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-parameterized PINNs work remains unknown.This paper presents the first convergence analysis of the overparameterized PINNs for the Laplace equations with Dirichlet boundary conditions.We demonstrate that the convergence rate can be controlled by the weight norm,regardless of the number of parameters in the network.
