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
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
Accelerated aging tests are widely used to rapidly evaluate the durability of materials,of which thermal-oxidative aging is the most common approach.To quantitatively predict the effects of multiple coupled factors,th...Accelerated aging tests are widely used to rapidly evaluate the durability of materials,of which thermal-oxidative aging is the most common approach.To quantitatively predict the effects of multiple coupled factors,this study takes polyamide66 reinforced with glass fiber(PA66-GF)as a model system and proposed a high-precision paradigm for coupled thermal-oxidative aging.By integrating Arrhenius-type reaction kinetics with oxygen diffusion,a predictive formula that holistically captures the nonlinear synergistic effects of multiple factors was developed,thereby overcoming the limitations of traditional single-variable models.A systematic evaluation of the stepwise improved formulas through nonlinear fitting showed that the coefficient of determination(R^(2))increased from 0.223 to 0.803,elucidating the fundamental reason why conventional approaches fail in quantitative prediction.These formulae were further embedded as physical constraints into a physics-informed neural network(PINN),which further enhanced the predictive performance,with the proposed formula achieving a peak R^(2)of 0.946.The results highlight that robust data fitting alone is insufficient;the decisive factor for the success of PINN lies in whether the embedded formula faithfully reflects the underlying physical mechanisms.When applied to polyamide 6 reinforced with glass fiber(PA6-GF),the Formula-constrained PINN maintained a high level of accuracy(R^(2)=0.916),demonstrating its strong cross-system generalizability.In summary,this work establishes a robust hybrid physics-machine learning framework that combines high accuracy with transferability for predicting the thermal-oxidative aging behavior of composite material systems.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
Fluid flow through fractured rock masses is a key process controlling the safety and performance of deep geoengineering systems,shaped by the complex interactions of thermal,hydraulic,mechanical and chemical(THMC)fiel...Fluid flow through fractured rock masses is a key process controlling the safety and performance of deep geoengineering systems,shaped by the complex interactions of thermal,hydraulic,mechanical and chemical(THMC)fields.This paper presents a systematic review of this subject with special emphasis on the multi-physics governing it.First,we elucidate the interdependent mechanisms and governing equations,highlighting the nonlinear,path-dependent,and evolving nature of the relationship between stress and permeability.Next,mainstream modeling approaches,including equivalent continuum,discrete fracture network(DFN),and dual-porosity/dual-permeability methods,are critically evaluated,and a strategy for model selection based on project scale and geological context is proposed accordingly.Moreover,experimental insights from single-fracture and triaxial flow studies are synthesized,revealing how effective stress,shear displacement,and fracture roughness control permeability evolution.In particular,the practical significance of THMC coupling is demonstrated through case studies on nuclear waste disposal,Enhanced Geothermal Systems,and tunneling projects.The reviewfurther explores AI-and machine learning-driven innovations,particularly physics-informed neural networks and hybrid modeling,which address limitations in computational efficiency,data scarcity,and physical consistency.Finally,persistent challenges,including multi-scale coupling,parameter uncertainty,and complex fracture network representation are identified and critically discussed while paying attention to future developments.展开更多
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.
基金financially supported by the National Natural Science Foundation of China(No.22473032)。
文摘Accelerated aging tests are widely used to rapidly evaluate the durability of materials,of which thermal-oxidative aging is the most common approach.To quantitatively predict the effects of multiple coupled factors,this study takes polyamide66 reinforced with glass fiber(PA66-GF)as a model system and proposed a high-precision paradigm for coupled thermal-oxidative aging.By integrating Arrhenius-type reaction kinetics with oxygen diffusion,a predictive formula that holistically captures the nonlinear synergistic effects of multiple factors was developed,thereby overcoming the limitations of traditional single-variable models.A systematic evaluation of the stepwise improved formulas through nonlinear fitting showed that the coefficient of determination(R^(2))increased from 0.223 to 0.803,elucidating the fundamental reason why conventional approaches fail in quantitative prediction.These formulae were further embedded as physical constraints into a physics-informed neural network(PINN),which further enhanced the predictive performance,with the proposed formula achieving a peak R^(2)of 0.946.The results highlight that robust data fitting alone is insufficient;the decisive factor for the success of PINN lies in whether the embedded formula faithfully reflects the underlying physical mechanisms.When applied to polyamide 6 reinforced with glass fiber(PA6-GF),the Formula-constrained PINN maintained a high level of accuracy(R^(2)=0.916),demonstrating its strong cross-system generalizability.In summary,this work establishes a robust hybrid physics-machine learning framework that combines high accuracy with transferability for predicting the thermal-oxidative aging behavior of composite material systems.
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.
文摘Fluid flow through fractured rock masses is a key process controlling the safety and performance of deep geoengineering systems,shaped by the complex interactions of thermal,hydraulic,mechanical and chemical(THMC)fields.This paper presents a systematic review of this subject with special emphasis on the multi-physics governing it.First,we elucidate the interdependent mechanisms and governing equations,highlighting the nonlinear,path-dependent,and evolving nature of the relationship between stress and permeability.Next,mainstream modeling approaches,including equivalent continuum,discrete fracture network(DFN),and dual-porosity/dual-permeability methods,are critically evaluated,and a strategy for model selection based on project scale and geological context is proposed accordingly.Moreover,experimental insights from single-fracture and triaxial flow studies are synthesized,revealing how effective stress,shear displacement,and fracture roughness control permeability evolution.In particular,the practical significance of THMC coupling is demonstrated through case studies on nuclear waste disposal,Enhanced Geothermal Systems,and tunneling projects.The reviewfurther explores AI-and machine learning-driven innovations,particularly physics-informed neural networks and hybrid modeling,which address limitations in computational efficiency,data scarcity,and physical consistency.Finally,persistent challenges,including multi-scale coupling,parameter uncertainty,and complex fracture network representation are identified and critically discussed while paying attention to future developments.
