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A Scalable Domain Decomposition Method for Ultra-Parallel Arterial Flow Simulations

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Communications in Computational Physics 2008年第10期4卷 1151-1169页

作者： Leopold Grinberg george em karniadakis Division of Applied Mathematics Brown UniversityProvidence 02912USA

Ultra-parallel flow simulations on hundreds of thousands of processors require new multi-level domain decomposition *** we present such a new two-level method that has features both of discontinuous and continuous Galerkin ***,at the coarse level the domain is subdivided into several big patches and within each patch a spectral element discretization(fine level)is *** interface conditions for the Navier-Stokes equations are developed to connect the patches,relaxing the C0 continuity and minimizing data transfer at the patch *** perform several 3D flow simulations of a benchmark problem and of arterial flows to evaluate the performance of the new method and investigate its accuracy.

关键词： Bioflows spectral elements discontinuous Galerkin parallel computing

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

Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients

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Communications in Computational Physics 2009年第2期5卷 793-820页

作者： Xiaoliang Wan george em karniadakis Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos(ME-gPC)method for elliptic equations with random coefficients for a moderate number(O(10))of random *** employ loworder(p≤3)polynomial chaos and refine the solution using adaptivity in the parametric *** first study the approximation error of ME-gPC and prove its *** subsequently generate local and global a posteriori error *** order to resolve the error equations efficiently,we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC *** on the a posteriori estimators,we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random *** results for convergence and efficiency are also presented.

关键词： Stochastic PDE a posteriori error estimate elliptic problems adaptive numerical methods uncertainty quantification

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

Analytical and Computational Studies of Correlations of Hydrodynamic Fluctuations in Shear Flow

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Communications in Computational Physics 2018年第1期23卷 93-117页

作者： Xin Bian Mingge Deng george em karniadakis Division of Applied Mathematics Brown UniversityProvidenceRhode Island 02912USA Lehrstuhl fur Aerodynamik und Stromungsmechanik Technische Universitat Munchen85748 GarchingGermany

We study correlations of hydrodynamic fluctuations in shear flow analytically and also by dissipative particle dynamics(DPD)*** hydrodynamic equations are linearized around the macroscopic velocity field and then solved by a perturbation method in Fourier-transformed *** autocorrelation functions(ACFs)fromthe analyticalmethod are comparedwith results obtained fromDPD simulations under the same shear-flow *** to a moderate shear rate,various ACFs from the two approaches agree with each other *** large shear rates,discrepancies between the two methods are observed,hence revealing strong additional coupling between different fluctuating variables,which is not considered in the analytical *** addition,the results at low and moderate shear rates can serve as benchmarks for developing multiscale algorithms for coupling of heterogeneous solvers,such as a hybrid simulation of molecular dynamics and fluctuating hydrodynamics solver,where thermal fluctuations are indispensable.

关键词： Hydrodynamic fluctuations correlation functions dissipative particle dynamics shear flow

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

Learning two-phase microstructure evolution using neural operators and autoencoder architectures

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npj Computational Materials 2022年第1期8卷 1804-1816页

作者： Vivek Oommen Khemraj Shukla Somdatta Goswami Rémi Dingreville george em karniadakis School of Engineering Brown UniversityProvidenceRIUSA Division of Applied Mathematics Brown UniversityProvidenceRIUSA Center for Integrated Nanotechnologies Sandia National LaboratoriesAlbuquerqueNM87185USA

Phase-field modeling is an effective but computationally expensive method for capturing the mesoscale morphological and microstructure evolution in ***,fast and generalizable surrogate models are needed to alleviate the cost of computationally taxing processes such as in optimization and design of *** intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model *** develop a framework that integrates a convolutional autoencoder architecture with a deep neural operator(DeepONet)to learn the dynamic evolution of a two-phase mixture and accelerate time-to-solution in predicting the microstructure *** utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent *** DeepONet is trained in the latent space,it can be used to replace the high-fidelity phase-field numerical solver in interpolation tasks or to accelerate the numerical solver in extrapolation tasks.

关键词： materials phase operators

Fractional Buffer Layers:Absorbing Boundary Conditions for Wave Propagation

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Communications in Computational Physics 2022年第2期31卷 331-369页

作者： Min Cai Ehsan Kharazmi Changpin Li george em karniadakis Department of Mathematics Shanghai University99 Shangda RoadShanghai 200444P.R.China Division of AppliedMathematics Brown University170 Hope StProvidenceRI 02906USA School of Engineering Brown University184 Hope StProvidenceRI 02912USA

We develop fractional buffer layers(FBLs)to absorb propagating waves without reflection in bounded *** formulation is based on variable-order spatial fractional *** select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the *** particular,we first design proper FBLs for the one-dimensional one-way and two-waywave ***,we extend our formulation to two-dimensional problems,wherewe introduce a consistent variable-order fractionalwave *** each case,we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in *** compare our results with a finely tuned perfectly matched layer(PML)method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular *** also demonstrate that our formulation is more robust and uses less number of equations.

关键词： Variable-order fractional derivatives FBL wave equation

Bi-Orthogonal fPINN:A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs

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Communications in Computational Physics 2023年第9期34卷 1133-1176页

作者： Lei Ma Rongxin Li Fanhai Zeng Ling Guo george em karniadakis Department of Mathematics Shanghai Normal UniversityShanghai 200234China School of Mathematics Shandong UniversityShandong 250100China Division of Applied Mathematics Brown UniversityProvidenceRI 02906USAand School of EngineeringBrown UniversityProvidenceRI 02906USA

Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal ***,inherent uncertainties arise naturally in real applications due to random forcing or unknown material *** models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and ***,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent ***,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak *** automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network *** weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue ***,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward *** demonstrate the effectiveness of the BO-fPINN method for different benchmark ***,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method *** then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial *** study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN *** results demonstrate the flexibility and efficiency

关键词： Scientific machine learning uncertainty quantification stochastic fractional differential equations PINNs inverse problems