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Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions

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Applied Mathematics and Mechanics(English Edition) 2023年第7期44卷 1069-1084页

作者： zhiping mao Xuhui MENG School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamen 361005Fujian ProvinceChina Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhan 430074China

We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this *** 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 *** 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 *** 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 *** 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 *** 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 *** 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 ***,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠ*** means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the ***,we also employ the similar adaptive sampling technique fo

关键词： physics-informed neural network(PINN) adaptive sampling high-dimension L-shape Poisson equation accuracy

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Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy

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Communications on Applied Mathematics and Computation 2019年第1期1卷 5-19页

作者： Mark Ainsworth zhiping mao Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the *** equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the *** study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard *** particular,we show that,analogous to the Cahn-Hilliard case,the solutions consist of regions in which the solution is a piecewise constant(whose value depends on the mass and the fractional order)separated by an interface whose width is independent of the mass and the fractional ***,if the average value of the initial data exceeds some threshold(which we determine explic让ly),then the solution will tend to a single constant steady state.

关键词： Fractional differential equation Non-local energy Well-posedness Fourier spectral method

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Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 876-902页

作者： Xuejuan Chen zhiping mao George Em Karniadakis School of Science Jimei UniversityXiamen 361021China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC *** then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff ***,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational ***,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.

关键词： Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models

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High Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows

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Communications in Computational Physics 2022年第10期32卷 1474-1509页

作者： Nan Zhang zhiping mao Tao Xiong School of Mathematical Sciences Xiamen UniversityXiamenFujian361005P.R.China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenFujian361005P.R.China

In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential *** fractional Laplacian is approximated by a Fourier-Galerkin spectral method,while the time evolution of the potential temperature is discretized by a high order conservative finite difference *** essentially non-oscillatory(WENO)reconstructions are also considered for *** to a low regularity of problems involving such a fractional Laplacian,especially in the critical or supercritical regime,directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity,*** smooth *** of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference *** results are performed to demonstrate the good performance of our proposed approach.

关键词： Finite difference WENO Fourier spectral fractional Laplacian surface quasigeostrophic flow

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Physics-informed neural networks(PINNs)for fluidmechanics:a review

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Acta Mechanica Sinica 2021年第12期37卷 1727-1738页

作者： Shengze Cai zhiping mao Zhicheng Wang Minglang Yin George Em Karniadakis Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences Xiamen University 361005 China Laboratory of Ocean Energy Utilization of Ministry of Education Dalian University of TechnologyDalian116024China School of Engineering Brown UniversityProvidenceRI 02912USA Center for Biomedical Engineering Brown UniversityProvidenceRI 02912USA

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 ***,solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer ***,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.

关键词： Physics-informed learning PINNs Inverse problems Supersonic flows Biomedical flows

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Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

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Communications on Applied Mathematics and Computation 2019年第4期1卷 597-619页

作者： zhiping mao Zhen Li George Em Karniadakis Division of Applied Mathematics Brown UniversityProvidenceRI02912USA Pacific Northwest National Laboratory RichlandWA 99354USA

Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large *** employ an agent-based model to describe the microscopic dynamics of each individual in a flock,and use a fractional partial differential equation(fPDE)to model the evolution of macroscopic quantities of *** macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic *** of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics,we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based *** demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking *** particular,a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual,while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale *** trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained *** can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian *** show in one-and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle *** proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models,and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.

关键词： Fractional PDEs Gaussian process Bayesian optimization Fractional Laplacian Conservation laws

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Hydrogel Fiber Actuators Prepared by Shell-Core Structure for High‑Performance Water/Light Dual Response

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Advanced Fiber Materials 2024年第6期6卷 1887-1897页

作者： Qianqian Wang Linping Zhang Yi Zhong Hong Xu zhiping mao Key Lab of Science and Technology of Eco‑Textile Ministry of EducationCollege of Chemistry and Chemical EngineeringDonghua UniversityShanghai 201620China Shanghai Belt and Road Joint Laboratory of Textile Intelligent Manufacturing Innovation Center for Textile Science and TechnologyDonghua UniversityShanghai 201620China

Spiral fibers with high energy storage and high output efficiency are highly desirable for soft robots and ***,it is still a great challenge to achieve spiral fibers with excellent water actuation performance,structural stability,and high scalability in a low-cost strategy.A coaxial spiral structure is reported for the fabrication of high-performance fiber *** developed shell-core helical fiber actuators were based on alginate/poly(ethylene glycol)acrylate shell and alginate/GO core with green and excellent *** to the high water-absorbing-swelling capacity and energy storage of the shell,the prepared spiral fibers are characterized by fast response,high energy output,and good repeatability of *** the other hand,the core endows the spiral fibers with the additional features of strong force retention and photothermal *** shell-core spiral structure promotes the output efficiency of the twisted fiber actuator with a large rotation(2500°/cm),untwisting speed(2250 rpm),and recovery speed(2700 rpm).In addition,the tertiary spiral structure based on TAPG fibers exhibits excellent humidity and photothermal response *** application of fibers to smart textiles enables efficient human epidermal thermal management.

关键词： Hydrogel fibers Shell-core structure Rapid response Large recovery speed Water/light dual response

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Thermally conductive poly(lactic acid)/boron nitride composites via regenerated cellulose assisted Pickering emulsion approach

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Journal of Materials Science & Technology 2022年第6期101卷 146-154页

作者： Yating Wang Hong Jin Jiajun Shen Bijia Wang Xueling Feng zhiping mao Yumei Zhang Xiaofeng Sui Key Lab of Science and Technology of Eco-Textile Ministry of EducationCollege of ChemistryChemical Engineering and BiotechnologyDonghua UniversityNo.2999North Renmin RoadSongjiang DistrictShanghai 201620China Innovation Center for Textile Science and Technology of DHU Donghua UniversityNo.2999North Renmin RoadSongjiang DistrictShanghai 201620China State Key Laboratory for Modification of Chemical Fibers and Polymer Materials College of Materials Science and EngineeringDonghua UniversityNo.2999North Renmin RoadSongjiang DistrictShanghai 201620China

Polymer-based thermally conductive composites have attracted tremendous interest in thermal management of ***,it remains challenging to achieve high thermal conductivity partly because the difficulty to obtain favorable distribution and orientation of conductive fillers within the polymer ***,networked boron nitride(BN)conductive pathway was realized within the poly(lactic acid)(PLA)matrix,via regenerated cellulose(RC)-assisted assembly of BN on Pickering emulsion interface based on the noncovalent interaction,followed by solvent evaporation and *** strong noncovalent interactions between BN and RC were found critical to enhance the wettability and stability of BN in aqueous media with a lowest mass ratio of 1:40 of RC and *** obtained PLA/BN composites feature a thermal conductivity of 1.06 W/(m K)at 28.4 wt%BN loading,representing an enhancement of 430%comparing to neat PLA,and the crystallinity of the composites could increase significantly from11.7%(neat PLA)to 43.7%.This simple,environmentally friendly and effective strategy could be easily extended for effective construction of thermally conductive composites.

关键词： Boron nitride Regenerated cellulose Emulsion Noncovalent interaction Thermal conductivity

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Polyoxometalate-loaded Nanocellulose Sponge as a Novel Catalyst for the Regioselective Hydroboration of Phenylacetylene

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Paper And Biomaterials 2019年第2期4卷 1-9页

作者： GuoQing Yang CongCong Zhang BiJia Wang zhiping mao Hong Xu Yi Zhong XueLing Feng XiaoFeng Sui Key Lab of Science and Technology of Eco-textile Ministry of EducationCollege of ChemistryChemical Engineering and BiotechnologyDonghua UniversityShanghai201620China Innovation Center for Textile Science and Technology of DHU Donghua UniversityShanghai201620China

In this work,an amino-modified cellulose nanofiber sponge was prepared and used as a support for polyoxometalate(POM)catalysts with a high loading *** transform infrared spectroscopy,thermogravimetric analysis,and energy-dispersive X-ray spectroscopy revealed that an Anderson-type POM,(NH4)4[CuMo6O18(OH)6]·5H2O was successfully immobilized on the sponge based on electrostatic *** analysis indicated that the POM-loaded sponge retained its porous structure and that the POM was homogeneously distributed on the sponge *** POM-loaded sponge exhibited excellent mechanical properties by recovering 79.9%of its original thickness following a 60%compression *** POM-loaded sponge was found to effectively catalyze the hydroboration of phenylacetylenes,yielding excellent conversion and regioselectivity of up to 96%and 99%,*** catalytic activity remained unchanged after five reuse *** findings represent a scalable strategy for immobilizing POMs on porous supports.

关键词： cellulose nanofiber polyoxometalates catalyst immobilization hydroboration heterogeneous catalyst

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Preparation and Characterization of Sepiolite Microfibers with High Aspect Ratio

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Journal of Donghua University(English Edition) 2022年第6期39卷 542-548页

作者： XU Tong HU Yelei QIAN Di ZHU Yuanzhao ZHONG Yi ZHANG Linping XU Hong mao zhiping Key Laboratory of Science&Technology of Eco-Textile Ministry of EducationDonghua UniversityShanghai 201620China National Dyeing and Finishing Engineering Technology Research Center Donghua UniversityShanghai 201620China National Manufacturing Innovation Center of Advanced Dyeing and Finishing Technology Taian 271000China Key Laboratory of Bioorganic Phosphorus Chemistry&Chemical Biology Ministry of EducationDepartment of ChemistryTsinghua UniversityBeijing 100084China Sino-German Engineering College Shanghai Technical Institute of Electronics&InformationShanghai 201411China

It is difficult to access exfoliated sepiolite(Sep)fibers with high aspect ratio from Sep *** traditional method used to purify Sep ore also reduces its aspect *** this study,impurities in the Sep ore were removed by acid activation followed by a cetyltrimethylammonium chloride(C16)treatment to organically modify the purified Sep by cation ***,the organically-modified Sep(O-Sep)was stripped and processed by an ultrasonic cell crusher to obtain Sep microfibers at a specific frequency for a given *** Sep samples had relatively high aspect ratio,compared with the Sep fibers gotten by traditional *** electron microscopy(SEM)and transmission electron microscopy(TEM)demonstrate the micro-morphology of exfoliated Sep samples in an intuitive ***,pure inorganic membrane prepared only with the exfoliated Sep fibers exhibited excellent flexibility,further demonstrating the excellent properties of Sep fibers with high aspect ratio.

关键词： sepiolite microfiber high aspect ratio inorganic membrane flexibility

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