Recent advances in deep neural networks have shed new light on physics,engineering,and scientific computing.Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots.The physicsi...Recent advances in deep neural networks have shed new light on physics,engineering,and scientific computing.Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots.The physicsinformedneural network(PINN)is currently the most general framework,which is more popular due to theconvenience of constructing NNs and excellent generalization ability.The automatic differentiation(AD)-basedPINN model is suitable for the homogeneous scientific problem;however,it is unclear how AD can enforce fluxcontinuity across boundaries between cells of different properties where spatial heterogeneity is represented bygrid cells with different physical properties.In this work,we propose a criss-cross physics-informed convolutionalneural network(CC-PINN)learning architecture,aiming to learn the solution of parametric PDEs with spatialheterogeneity of physical properties.To achieve the seamless enforcement of flux continuity and integration ofphysicalmeaning into CNN,a predefined 2D convolutional layer is proposed to accurately express transmissibilitybetween adjacent cells.The efficacy of the proposedmethodwas evaluated through predictions of several petroleumreservoir problems with spatial heterogeneity and compared against state-of-the-art(PINN)through numericalanalysis as a benchmark,which demonstrated the superiority of the proposed method over the PINN.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
Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or inte...Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or interior data.However,the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported.The present paper proposes an artificial viscosity(AV)-based PINN for solving the forward and inverse flow problems.Specifically,the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics(CFD)to stabilize the simulation of flow at high Reynolds numbers.The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re=1000 and the inverse problem derived from two-dimensional film boiling.The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.展开更多
The parametric surrogate models for partial differential equations(PDEs)are a necessary component for many applications in computational sciences,and the convolutional neural networks(CNNs)have proven to be an excelle...The parametric surrogate models for partial differential equations(PDEs)are a necessary component for many applications in computational sciences,and the convolutional neural networks(CNNs)have proven to be an excellent tool to generate these surrogates when parametric fields are present.CNNs are commonly trained on labeled data based on one-to-one sets of parameter-input and PDE-output fields.Recently,residual-based deep convolutional physics-informed neural network(DCPINN)solvers for parametric PDEs have been proposed to build surrogates without the need for labeled data.These allow for the generation of surrogates without an expensive offline-phase.In this work,we present an alternative formulation termed deep convolutional Ritz method(DCRM)as a parametric PDE solver.The approach is based on the minimization of energy functionals,which lowers the order of the differential operators compared to residualbased methods.Based on studies involving the Poisson equation with a spatially parameterized source term and boundary conditions,we find that CNNs trained on labeled data outperform DCPINNs in convergence speed and generalization abilities.The surrogates generated from the DCRM,however,converge significantly faster than their DCPINN counterparts,and prove to generalize faster and better than the surrogates obtained from both CNNs trained on labeled data and DCPINNs.This hints that the DCRM could make PDE solution surrogates trained without labeled data possibly.展开更多
Physics-informed deep learning has recently emerged as an effective tool for leveraging both observational data and available physical laws.Physics-informed neural networks(PINNs)and deep operator networks(DeepONets)a...Physics-informed deep learning has recently emerged as an effective tool for leveraging both observational data and available physical laws.Physics-informed neural networks(PINNs)and deep operator networks(DeepONets)are two such models.The former encodes the physical laws via the automatic differentiation,while the latter learns the hidden physics from data.Generally,the noisy and limited observational data as well as the over-parameterization in neural networks(NNs)result in uncertainty in predictions from deep learning models.In paper“MENG,X.,YANG,L.,MAO,Z.,FERRANDIS,J.D.,and KARNIADAKIS,G.E.Learning functional priors and posteriors from data and physics.Journal of Computational Physics,457,111073(2022)”,a Bayesian framework based on the generative adversarial networks(GANs)has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets.Specifically,the proposed approach in“MENG,X.,YANG,L.,MAO,Z.,FERRANDIS,J.D.,and KARNIADAKIS,G.E.Learning functional priors and posteriors from data and physics.Journal of Computational Physics,457,111073(2022)”has two stages:(i)prior learning,and(ii)posterior estimation.At the first stage,the GANs are utilized to learn a functional prior either from a prescribed function distribution,e.g.,the Gaussian process,or from historical data and available physics.At the second stage,the Hamiltonian Monte Carlo(HMC)method is utilized to estimate the posterior in the latent space of GANs.However,the vanilla HMC does not support the mini-batch training,which limits its applications in problems with big data.In the present work,we propose to use the normalizing flow(NF)models in the context of variational inference(VI),which naturally enables the mini-batch training,as the alternative to HMC for posterior estimation in the latent space of GANs.A series of numerical experiments,including a nonlinear differential equation problem and a 100-dimensional(100D)Darcy problem,are conducted to demonstrate that the NFs with full-/mini-batch training are able to achieve similar accuracy as the“gold rule”HMC.Moreover,the mini-batch training of NF makes it a promising tool for quantifying uncertainty in solving the high-dimensional partial differential equation(PDE)problems with big data.展开更多
Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems,whose basic concept is to embed physical laws to constrain/inform neural networks,with the need of l...Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems,whose basic concept is to embed physical laws to constrain/inform neural networks,with the need of less data for training a reliable model.This can be achieved by incorporating the residual of physics equations into the loss function.Through minimizing the loss function,the network could approximate the solution.In this paper,we propose a mixed-variable scheme of physics-informed neural network(PINN)for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers.A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy.The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions.Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.展开更多
Prognosis of bearing is critical to improve the safety,reliability,and availability of machinery systems,which provides the health condition assessment and determines how long the machine would work before failure occ...Prognosis of bearing is critical to improve the safety,reliability,and availability of machinery systems,which provides the health condition assessment and determines how long the machine would work before failure occurs by predicting the remaining useful life(RUL).In order to overcome the drawback of pure data-driven methods and predict RUL accurately,a novel physics-informed deep neural network,named degradation consistency recurrent neural network,is proposed for RUL prediction by integrating the natural degradation knowledge of mechanical components.The degradation is monotonic over the whole life of bearings,which is characterized by temperature signals.To incorporate the knowledge of monotonic degradation,a positive increment recurrence relationship is introduced to keep the monotonicity.Thus,the proposed model is relatively well understood and capable to keep the learning process consistent with physical degradation.The effectiveness and merit of the RUL prediction using the proposed method are demonstrated through vibration signals collected from a set of run-to-failure tests.展开更多
基金the National Natural Science Foundation of China(No.52274048)Beijing Natural Science Foundation(No.3222037)+1 种基金the CNPC 14th Five-Year Perspective Fundamental Research Project(No.2021DJ2104)the Science Foundation of China University of Petroleum,Beijing(No.2462021YXZZ010).
文摘Recent advances in deep neural networks have shed new light on physics,engineering,and scientific computing.Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots.The physicsinformedneural network(PINN)is currently the most general framework,which is more popular due to theconvenience of constructing NNs and excellent generalization ability.The automatic differentiation(AD)-basedPINN model is suitable for the homogeneous scientific problem;however,it is unclear how AD can enforce fluxcontinuity across boundaries between cells of different properties where spatial heterogeneity is represented bygrid cells with different physical properties.In this work,we propose a criss-cross physics-informed convolutionalneural network(CC-PINN)learning architecture,aiming to learn the solution of parametric PDEs with spatialheterogeneity of physical properties.To achieve the seamless enforcement of flux continuity and integration ofphysicalmeaning into CNN,a predefined 2D convolutional layer is proposed to accurately express transmissibilitybetween adjacent cells.The efficacy of the proposedmethodwas evaluated through predictions of several petroleumreservoir problems with spatial heterogeneity and compared against state-of-the-art(PINN)through numericalanalysis as a benchmark,which demonstrated the superiority of the proposed method over the PINN.
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 Fundamental Research Funds for the Central Universities of China(No.DUT21RC(3)063)the National Natural Science Foundation of China(No.51720105007)the Baidu Foundation(No.ghfund202202014542)。
文摘Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or interior data.However,the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported.The present paper proposes an artificial viscosity(AV)-based PINN for solving the forward and inverse flow problems.Specifically,the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics(CFD)to stabilize the simulation of flow at high Reynolds numbers.The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re=1000 and the inverse problem derived from two-dimensional film boiling.The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.
基金supported by the Laboratory Directed Research and Development Program at Sandia National Laboratories(No.218328)。
文摘The parametric surrogate models for partial differential equations(PDEs)are a necessary component for many applications in computational sciences,and the convolutional neural networks(CNNs)have proven to be an excellent tool to generate these surrogates when parametric fields are present.CNNs are commonly trained on labeled data based on one-to-one sets of parameter-input and PDE-output fields.Recently,residual-based deep convolutional physics-informed neural network(DCPINN)solvers for parametric PDEs have been proposed to build surrogates without the need for labeled data.These allow for the generation of surrogates without an expensive offline-phase.In this work,we present an alternative formulation termed deep convolutional Ritz method(DCRM)as a parametric PDE solver.The approach is based on the minimization of energy functionals,which lowers the order of the differential operators compared to residualbased methods.Based on studies involving the Poisson equation with a spatially parameterized source term and boundary conditions,we find that CNNs trained on labeled data outperform DCPINNs in convergence speed and generalization abilities.The surrogates generated from the DCRM,however,converge significantly faster than their DCPINN counterparts,and prove to generalize faster and better than the surrogates obtained from both CNNs trained on labeled data and DCPINNs.This hints that the DCRM could make PDE solution surrogates trained without labeled data possibly.
基金Project supported by the National Natural Science Foundation of China(No.12201229)。
文摘Physics-informed deep learning has recently emerged as an effective tool for leveraging both observational data and available physical laws.Physics-informed neural networks(PINNs)and deep operator networks(DeepONets)are two such models.The former encodes the physical laws via the automatic differentiation,while the latter learns the hidden physics from data.Generally,the noisy and limited observational data as well as the over-parameterization in neural networks(NNs)result in uncertainty in predictions from deep learning models.In paper“MENG,X.,YANG,L.,MAO,Z.,FERRANDIS,J.D.,and KARNIADAKIS,G.E.Learning functional priors and posteriors from data and physics.Journal of Computational Physics,457,111073(2022)”,a Bayesian framework based on the generative adversarial networks(GANs)has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets.Specifically,the proposed approach in“MENG,X.,YANG,L.,MAO,Z.,FERRANDIS,J.D.,and KARNIADAKIS,G.E.Learning functional priors and posteriors from data and physics.Journal of Computational Physics,457,111073(2022)”has two stages:(i)prior learning,and(ii)posterior estimation.At the first stage,the GANs are utilized to learn a functional prior either from a prescribed function distribution,e.g.,the Gaussian process,or from historical data and available physics.At the second stage,the Hamiltonian Monte Carlo(HMC)method is utilized to estimate the posterior in the latent space of GANs.However,the vanilla HMC does not support the mini-batch training,which limits its applications in problems with big data.In the present work,we propose to use the normalizing flow(NF)models in the context of variational inference(VI),which naturally enables the mini-batch training,as the alternative to HMC for posterior estimation in the latent space of GANs.A series of numerical experiments,including a nonlinear differential equation problem and a 100-dimensional(100D)Darcy problem,are conducted to demonstrate that the NFs with full-/mini-batch training are able to achieve similar accuracy as the“gold rule”HMC.Moreover,the mini-batch training of NF makes it a promising tool for quantifying uncertainty in solving the high-dimensional partial differential equation(PDE)problems with big data.
文摘Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems,whose basic concept is to embed physical laws to constrain/inform neural networks,with the need of less data for training a reliable model.This can be achieved by incorporating the residual of physics equations into the loss function.Through minimizing the loss function,the network could approximate the solution.In this paper,we propose a mixed-variable scheme of physics-informed neural network(PINN)for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers.A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy.The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions.Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.
基金support in part by China Postdoctoral Science Foundation (No.2021M702634)National Science Foundation of China (No.52175116).
文摘Prognosis of bearing is critical to improve the safety,reliability,and availability of machinery systems,which provides the health condition assessment and determines how long the machine would work before failure occurs by predicting the remaining useful life(RUL).In order to overcome the drawback of pure data-driven methods and predict RUL accurately,a novel physics-informed deep neural network,named degradation consistency recurrent neural network,is proposed for RUL prediction by integrating the natural degradation knowledge of mechanical components.The degradation is monotonic over the whole life of bearings,which is characterized by temperature signals.To incorporate the knowledge of monotonic degradation,a positive increment recurrence relationship is introduced to keep the monotonicity.Thus,the proposed model is relatively well understood and capable to keep the learning process consistent with physical degradation.The effectiveness and merit of the RUL prediction using the proposed method are demonstrated through vibration signals collected from a set of run-to-failure tests.
基金supported by the National Natural Science Foundation of China(Grant No.91852117)the foundation of National Key Laboratory of Science and Technology on Aerodynamic Design and Research(Grant No.614220121040106)Shanghai Rising-Star Program(Grant No.19QC1400200)。
