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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 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

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

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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 method.We 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 problems.Furthermore,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 cost.Moreover,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 temperature.The 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 scheme.Weighted essentially non-oscillatory(WENO)reconstructions are also considered for comparison.Due 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,e.***.around smooth extrema.Instead of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference discretizations.Numerical 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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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 electronics.However,it remains challenging to achieve high thermal conductivity partly because the difficulty to obtain favorable distribution and orientation of conductive fillers within the polymer matrix.Herein,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 hot-compressing.The 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 BN.The 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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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 solution.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 energy.We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case.In 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 derivative.However,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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分散染料在超临界CO_(2)流体染色聚酯纤维中的扩散行为

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纺织学报 2024年第2期45卷 134-141页

作者：范博吴伟王健徐红毛志平东华大学生态纺织教育部重点实验室上海201620 东华大学化学与化工学院上海201620 青岛即发集团股份有限公司山东青岛266000 东华大学纺织科技创新中心上海201620 东华大学国家染整工程技术研究中心上海201620

为深入研究分散染料的扩散性能在超临界CO_(2)染色过程中影响染色条件及染色质量的理论机制,使用共聚焦拉曼显微镜研究了超临界CO_(2)流体染色在不同工况条件(100~140℃、21~27 MPa、5~90 min)下分散染料(分散红167、分散橙30、分散蓝79... 详细信息

为深入研究分散染料的扩散性能在超临界CO_(2)染色过程中影响染色条件及染色质量的理论机制,使用共聚焦拉曼显微镜研究了超临界CO_(2)流体染色在不同工况条件(100~140℃、21~27 MPa、5~90 min)下分散染料(分散红167、分散橙30、分散蓝79)在涤纶中的扩散行为。通过拉曼深度成像图获得了染料在纤维径向分布的数据,计算扩散系数。结果显示:分散染料在染色初期(5 min内)已在纤维内分布均匀,且随染色时间、染色压力的增加,染料在纤维内部均匀上染。对比染色后纤维径向的拉曼光谱数据与纱线上染量数据,验证了拉曼光谱数据的有效性。经计算分散染料中分散橙30的扩散系数最高(9.744×10^(-15) m^(2)/s)。表明拉曼技术用于分析单纤维内染料扩散有着巨大的优势,反应快速,制样简单,同时可在无损纤维的前提下对涤纶内部的染料分布进行测量。

关键词：分散染料聚酯纤维超临界CO_(2)流体共聚焦拉曼显微镜扩散

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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 scale.We 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 interest.The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model.Instead 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 simulations.We 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 dynamics.In 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 quantities.The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally.They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization.We 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 method.The 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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5-(二甲氨基)-2-甲基-5-氧戊酸甲酯在超临界二氧化碳流体染色中的应用

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纺织学报 2024年第1期45卷 120-127页

作者：葛怀富吴伟王健徐红毛志平东华大学生态纺织教育部重点实验室上海201620 东华大学化学与化工学院上海201620 青岛即发集团股份有限公司山东青岛266000 东华大学纺织科技创新中心上海201620 东华大学国家染整工程技术研究中心上海201620

为提高部分分散染料在超临界二氧化碳流体(ScCO 2)中上染率,补全ScCO 2分散染料染色色谱,以5-(二甲氨基)-2-甲基-5-氧戊酸甲酯作为分散染料ScCO 2染色助溶剂,通过实验探究了染料与助溶剂质量比、染色温度、染色压力对分散染料上染率的影... 详细信息

为提高部分分散染料在超临界二氧化碳流体(ScCO 2)中上染率,补全ScCO 2分散染料染色色谱,以5-(二甲氨基)-2-甲基-5-氧戊酸甲酯作为分散染料ScCO 2染色助溶剂,通过实验探究了染料与助溶剂质量比、染色温度、染色压力对分散染料上染率的影响,并对染色后涤纶纱线的力学性能、耐摩擦色牢度、耐日晒色牢度以及耐皂洗色牢度进行表征。结果表明:纱线得色深度明显提高,且未对纱线色相、饱和度产生明显影响;分散黄163、分散蓝60染料滤饼上染率分别为21.61%、34.25%;加入助溶剂后,分散黄163、分散蓝60的ScCO 2染色上染率分别可达57.62%、70.97%;染色纱线得色深度显著提高,染色后纱线的K/S值分别由4.5、5.9提升至12.0、12.2;最佳染色工艺为:染料与助溶剂质量比1∶20,染色温度130℃,染色压力27 MPa;加入助溶剂后染色纱线的力学性能、色牢度符合生产需求。

关键词：分散染料超临界二氧化碳流体助溶剂上染率涤纶纱线

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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 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.

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

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柔性海泡石基气凝胶的制备及其保温隔热性能的研究

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化工新型材料 2024年第2期52卷 175-180页

作者：胡叶蕾续通毛志平徐红东华大学化学与化工学院生态纺织教育部重点实验室上海201620 上海电子信息技术学院中德工程学院上海201620 山东中康国创先进印染技术研究院有限公司国家先进印染技术创新中心泰安271000

以海泡石为基质,聚乙烯醇为有机增强相,硅烷偶联剂KH-560和聚乙烯亚胺为交联剂,经冷冻干燥制备了具有优异保温隔热性能的柔性海泡石基气凝胶。其导热系数低至0.0374W/(m·K),轴向在经过80%的压缩形变后能达到100%回弹,径向在经过50%的... 详细信息

以海泡石为基质,聚乙烯醇为有机增强相,硅烷偶联剂KH-560和聚乙烯亚胺为交联剂,经冷冻干燥制备了具有优异保温隔热性能的柔性海泡石基气凝胶。其导热系数低至0.0374W/(m·K),轴向在经过80%的压缩形变后能达到100%回弹,径向在经过50%的压缩形变后也能达到100%回弹。该气凝胶有望作为保温隔热材料应用于建筑、隔热帐篷等领域。

关键词：海泡石基气凝胶热稳定性力学性能保温隔热

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