Edit Profile (opens in new tab) Karniadakis, George Em Compute Distance To: Compute Author ID: karniadakis.george-em Published as: Karniadakis, George Em; Karniadakis, G. E.; Karniadakis, George E.; Karniadakis, George Em.; Karniadakis, George; Karniadakis, George E. M.; Em Karniadakis, George more...less Homepage: https://www.brown.edu/research/projects/crunch/george-karniadakis External Links: MGP · ORCID · Wikidata · Google Scholar · dblp · GND · IdRef Documents Indexed: 326 Publications since 1986, including 7 Books 6 Contributions as Editor Co-Authors: 222 Co-Authors with 326 Joint Publications 6,456 Co-Co-Authors all top 5 Co-Authors 6 single-authored 16 Mao, Zhiping 16 Venturi, Daniele 16 Wan, Xiaoliang 16 Zhang, Zhongqiang 14 Sherwin, Spencer J. 14 Zayernouri, Mohsen 13 Li, Zhen 12 Meng, Xuhui 12 Perdikaris, Paris G. 12 Xiu, Dongbin 12 Zeng, Fanhai 11 Triantafyllou, Michael S. 10 Lu, Lu 10 Raissi, Maziar 10 Wang, Zhi Cheng 9 Caswell, Bruce 9 Kirby, Robert M. II 9 Lucor, Didier 9 Song, Fangying 9 Su, Chau-Hsing 8 Dong, Suchuan 8 Jagtap, Ameya D. 8 Lin, Guang 8 Maxey, Martin R. 8 Yu, Yue 7 Deng, Mingge 7 Evangelinos, Constantinos 7 Grinberg, Leopold 7 Lomtev, Igor 7 Orszag, Steven Alan 7 Pang, Guofei 7 Rozovskii, Boris L. 6 Cho, Heyrim 6 Choi, Minseok 6 Fedosov, Dmitry A. 6 Giannakouros, John G. 6 Kharazmi, Ehsan 6 Sirisup, Sirod 6 Zhang, Dongkun 5 Babaee, Hessam 5 Bian, Xin 5 Cai, Wei 5 Constantinides, Yiannis 5 Guo, Ling 5 Henderson, Ronald D. 5 Ma, Xia 5 Sidilkover, David 5 Warburton, Timothy C. E. 5 Yin, Minglang 5 Zheng, Xiaoning 4 Baek, Hyoungsu 4 Bittencourt, Marco Lúcio 4 Cai, Shengze 4 Cao, Wanrong 4 Kawaguchi, Kenji 4 Lei, Huan 4 Patera, Anthony T. 4 Sapsis, Themistoklis P. 4 Symeonidis, Vasileios 4 Tang, Yuhang 4 Xu, Jin 4 Yang, Xiu 4 Zaki, Tamer A. 3 Ainsworth, Mark 3 Beskok, Ali 3 Chryssostomidis, Chryssostomos 3 Foo, Jasmine 3 Goswami, Somdatta 3 Gulian, Mamikon 3 Kevrekidis, Ioannis George 3 Kim, Changho 3 Rockwell, Donald 3 Shu, Chi-Wang 3 Tang, Yifa 3 Tretyakov, Michael V. 3 Triantafyllou, George S. 3 Yazdani, Alireza 3 Zheng, Mengdi 2 Batcho, Paul F. 2 Blumers, Ansel L. 2 Bourguet, Rémi 2 Burrage, Kevin 2 Chen, Xiaoli 2 Clark di Leoni, Patricio 2 Cockburn, Bernardo 2 Darbon, Jerome 2 D’Elia, Marta 2 Du, Yiqing 2 Duan, Jinqiao 2 Fan, Dixia 2 Jiang, Xiaoyun 2 Jin, Pengzhan 2 Kaiktsis, Lambros 2 Karamanos, G.-S. 2 Li, Xuejin 2 Lischke, Anna 2 Luo, Xian 2 Meneveau, Charles 2 Newman, David J. 2 Ouyang, Jie ...and 122 more Co-Authors all top 5 Serials 101 Journal of Computational Physics 42 Computer Methods in Applied Mechanics and Engineering 40 SIAM Journal on Scientific Computing 31 Journal of Fluid Mechanics 8 Communications in Computational Physics 7 International Journal for Numerical Methods in Engineering 7 Journal of Scientific Computing 6 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 4 SIAM Journal on Numerical Analysis 3 International Journal for Numerical Methods in Fluids 3 Applied Numerical Mathematics 3 Computational Mechanics 3 Fractional Calculus & Applied Analysis 3 Numerical Mathematics and Scientific Computation 2 Computers and Fluids 2 International Journal of Heat and Mass Transfer 2 Neural Networks 2 Proceedings of the National Academy of Sciences of the United States of America 2 European Journal of Mechanics. B. Fluids 2 Multiscale Modeling & Simulation 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Computers & Mathematics with Applications 1 Computers and Structures 1 Journal of Engineering Mathematics 1 Journal of Statistical Physics 1 Physics of Fluids, A 1 Fluid Dynamics Research 1 Theoretical and Computational Fluid Dynamics 1 Chaos, Solitons and Fractals 1 Bulletin of the Polish Academy of Sciences. Technical Sciences 1 Numerische Mathematik 1 Physica D 1 European Journal of Applied Mathematics 1 Journal of Non-Newtonian Fluid Mechanics 1 SIAM Review 1 Physics of Fluids 1 Electronic Journal of Differential Equations (EJDE) 1 Parallel Algorithms and Applications 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Flow, Turbulence and Combustion 1 HERMIS-\(\mu\pi\). Hellenic European Research on Mathematics and Informatics Science 1 Applied Mathematical Sciences 1 Interdisciplinary Applied Mathematics 1 Lecture Notes in Computational Science and Engineering 1 Science 1 International Journal for Uncertainty Quantification 1 Journal of Theoretical Biology 1 Transactions of Mathematics and its Applications 1 Handbook of Fractional Calculus with Applications 1 Communications on Applied Mathematics and Computation all top 5 Fields 176 Fluid mechanics (76-XX) 151 Numerical analysis (65-XX) 80 Partial differential equations (35-XX) 56 Probability theory and stochastic processes (60-XX) 46 Computer science (68-XX) 38 Mechanics of deformable solids (74-XX) 25 Ordinary differential equations (34-XX) 22 Biology and other natural sciences (92-XX) 19 Statistics (62-XX) 11 Statistical mechanics, structure of matter (82-XX) 9 Dynamical systems and ergodic theory (37-XX) 9 Approximations and expansions (41-XX) 9 Geophysics (86-XX) 7 Real functions (26-XX) 6 General and overarching topics; collections (00-XX) 6 Classical thermodynamics, heat transfer (80-XX) 4 Mechanics of particles and systems (70-XX) 4 Optics, electromagnetic theory (78-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Special functions (33-XX) 1 Integral equations (45-XX) 1 Convex and discrete geometry (52-XX) 1 Quantum theory (81-XX) Publications by Year all cited Publications top 5 cited Publications Citations contained in zbMATH Open 302 Publications have been cited 9,815 times in 5,497 Documents Cited by ▼ Year ▼ The Wiener–Askey polynomial chaos for stochastic differential equations. Zbl 1014.65004Xiu, Dongbin; Karniadakis, George Em 847 2002 Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Zbl 1415.68175Raissi, M.; Perdikaris, P.; Karniadakis, G. E. 491 2019 High-order splitting methods for the incompressible Navier-Stokes equations. Zbl 0738.76050Karniadakis, George Em; Israeli, Moshe; Orszag, Steven A. 401 1991 Spectral/\(hp\) element methods for computational fluid dynamics. 2nd ed. Zbl 1116.76002Karniadakis, George Em; Sherwin, Spencer J. 330 2005 The development of discontinuous Galerkin methods. Zbl 0989.76045Cockburn, Bernardo; Karniadakis, George E.; Shu, Chi-Wang 296 2000 Modeling uncertainty in flow simulations via generalized polynomial chaos. Zbl 1047.76111Xiu, Dongbin; Karniadakis, George Em 280 2003 Spectral/hp element methods for CFD. Zbl 0954.76001Karniadakis, George Em; Sherwin, Spencer J. 265 1999 An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. Zbl 1078.65008Wan, Xiaoliang; Karniadakis, George Em 180 2005 Microflows and nanoflows. Fundamentals and simulation. Foreword by Chih-Ming Ho. Zbl 1115.76003Karniadakis, George; Beskok, Ali; Narayan, Aluru 162 2005 Multi-element generalized polynomial chaos for arbitrary probability measures. Zbl 1128.65009Wan, Xiaoliang; Karniadakis, George Em 155 2006 Hidden physics models: machine learning of nonlinear partial differential equations. Zbl 1381.68248Raissi, Maziar; Karniadakis, George Em 143 2018 Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Zbl 1016.65001Xiu, Dongbin; Karniadakis, George Em 142 2002 Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. Zbl 1349.34095Zayernouri, Mohsen; Karniadakis, George Em 142 2013 Fractional spectral collocation method. Zbl 1294.65097Zayernouri, Mohsen; Karniadakis, George Em 131 2014 Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Zbl 0746.76021Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E.; Orszag, S. A. 125 1991 Spectral/\(hp\) element methods for computational fluid dynamics. Reprint of the 2nd hardback ed. (2005). Zbl 1256.76003Karniadakis, George Em; Sherwin, Spencer J. 111 2013 Machine learning of linear differential equations using Gaussian processes. Zbl 1380.68339Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 97 2017 A spectral vanishing viscosity method for large eddy simulations. Zbl 0984.76036Karamanos, G. S.; Karniadakis, G. E. 97 2000 Inferring solutions of differential equations using noisy multi-fidelity data. Zbl 1382.65229Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 95 2017 Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Zbl 1349.65531Zayernouri, Mohsen; Karniadakis, George Em 95 2015 Exponentially accurate spectral and spectral element methods for fractional ODEs. Zbl 1349.65257Zayernouri, Mohsen; Karniadakis, George Em 93 2014 A semi-Lagrangian high-order method for Navier-Stokes equations. Zbl 1028.76026Xiu, Dongbin; Karniadakis, George Em 90 2001 Second-order approximations for variable order fractional derivatives: algorithms and applications. Zbl 1349.65092Zhao, Xuan; Sun, Zhi-zhong; Karniadakis, George Em 88 2015 DeepXDE: a deep learning library for solving differential equations. Zbl 1459.65002Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em 87 2021 The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications. Zbl 1153.65008Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em 81 2008 A spectral viscosity method for correcting the long-term behavior of POD models. Zbl 1136.76412Sirisup, S.; Karniadakis, G. E. 79 2004 Multi-element probabilistic collocation method in high dimensions. Zbl 1181.65014Foo, Jasmine; Karniadakis, George Em 77 2010 A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. Zbl 1339.65197Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em 74 2015 A triangular spectral element method; applications to the incompressible Navier-Stokes equations. Zbl 1075.76621Sherwin, S. J.; Karniadakis, G. E. 74 1995 What is the fractional Laplacian? A comparative review with new results. Zbl 1453.35179Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaoning; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em 72 2020 Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. Zbl 0754.76043Karniadakis, George Em; Triantafyllou, George S. 70 1992 FPINNs: fractional physics-informed neural networks. Zbl 1420.35459Pang, Guofei; Lu, Lu; Karniadakis, George Em 68 2019 A low-dimensional model for simulating three-dimensional cylinder flow. Zbl 1001.76043Ma, Xia; Karniadakis, George Em 68 2002 A unified Petrov-Galerkin spectral method for fractional PDEs. Zbl 1425.65127Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em. 66 2015 Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. Zbl 1386.65030Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 64 2018 Long-term behavior of polynomial chaos in stochastic flow simulations. Zbl 1123.76058Wan, Xiaoliang; Karniadakis, George Em 62 2006 A new triangular and tetrahedral basis for high-order \((hp)\) finite element methods. Zbl 0837.73075Sherwin, Spencer J.; Karniadakis, George Em. 62 1995 Physics-informed neural networks for high-speed flows. Zbl 1442.76092Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em 58 2020 De-aliasing on non-uniform grids: algorithms and applications. Zbl 1161.76534Kirby, Robert M.; Karniadakis, George Em 57 2003 Discontinuous spectral element methods for time- and space-fractional advection equations. Zbl 1304.35757Zayernouri, Mohsen; Karniadakis, George Em 57 2014 Discontinuous Galerkin methods. Theory, computation and applications. 1st international symposium on DGM, Newport, RI, USA, May 24–26, 1999. Zbl 0935.00043 54 2000 Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Zbl 1442.92002Jagtap, Ameya D.; Kharazmi, Ehsan; Karniadakis, George Em 50 2020 Dynamics and low-dimensionality of a turbulent near wake. Zbl 0987.76041Ma, X.; Karamanos, G.-S.; Karniadakis, G. E. 50 2000 Reweighted \(\ell_1\) minimization method for stochastic elliptic differential equations. Zbl 1349.60113Yang, Xiu; Karniadakis, George Em 49 2013 Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Zbl 1478.76057Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em. 49 2020 Frequency selection and asymptotic states in laminar wakes. Zbl 0659.76043Karniadakis, George Em; Triantafyllou, George S. 48 1989 Time-dependent generalized polynomial chaos. Zbl 1201.65216Gerritsma, Marc; van der Steen, Jan-Bart; Vos, Peter; Karniadakis, George 48 2010 A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. Zbl 1422.65428Mao, Zhiping; Karniadakis, George Em 47 2018 Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. Zbl 1408.76428Yang, Xiu; Choi, Minseok; Lin, Guang; Karniadakis, George Em 46 2012 Micro flows. Fundamentals and simulation. Zbl 0998.76002Karniadakis, George Em; Beskok, Ali 46 2002 A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. Zbl 0956.76046Lomtev, I.; Kirby, R. M.; Karniadakis, G. E. 45 1999 A new stochastic approach to transient heat conduction modeling with uncertainty. Zbl 1038.80003Xiu, Dongbin; Karniadakis, George Em 45 2003 Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step. Zbl 0728.76057Kaiktsis, Lambros; Karniadakis, George Em; Orszag, Steven A. 44 1991 Tempered fractional Sturm-Liouville eigenproblems. Zbl 1323.34012Zayernouri, Mohsen; Ainsworth, Mark; Karniadakis, George Em 42 2015 Tetrahedral spectral elements for CFD. Zbl 0857.76044Sherwin, S. J.; Karniadakis, G. E. 42 1995 Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Zbl 1453.68165Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em 41 2020 Beyond Wiener-Askey expansions: handling arbitrary PDFs. Zbl 1102.65006Wan, Xiaoliang; Karniadakis, George Em 41 2006 Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Zbl 07419158Jagtap, Ameya D.; Karniadakis, George Em 41 2020 Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Zbl 1454.65008Zhang, Dongkun; Lu, Lu; Guo, Ling; Karniadakis, George Em 40 2019 A combined direct numerical simulation–particle image velocimetry study of the turbulent near wake. Zbl 1177.76156Dong, S.; Karniadakis, G. E.; Ekmekci, A.; Rockwell, D. 40 2006 A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. Zbl 1431.65193Zeng, Fanhai; Mao, Zhiping; Karniadakis, George Em 40 2017 A discontinuous Galerkin method for the viscous MHD equations. Zbl 0954.76051Warburton, T. C.; Karniadakis, G. E. 40 1999 A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains. Zbl 1349.76569Dong, S.; Karniadakis, G. E.; Chryssostomidis, C. 40 2014 hp-VPINNs: variational physics-informed neural networks with domain decomposition. Zbl 07338008Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M. 39 2021 Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. Zbl 1355.65104Cao, Wanrong; Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em 38 2016 Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Zbl 1439.65081Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em 38 2017 A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Zbl 1423.76102Song, Fangying; Xu, Chuanju; Karniadakis, George Em 38 2016 Petrov-Galerkin and spectral collocation methods for distributed order differential equations. Zbl 1367.65113Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em 38 2017 Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. Zbl 1352.65278Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em 38 2016 Spectral polynomial chaos solutions of the stochastic advection equation. Zbl 1001.76084Jardak, M.; Su, C.-H.; Karniadakis, G. E. 37 2002 Spectral and discontinuous spectral element methods for fractional delay equations. Zbl 1314.34159Zayernouri, Mohsen; Cao, Wanrong; Zhang, Zhongqiang; Karniadakis, George Em 36 2014 Numerical methods for stochastic partial differential equations with white noise. Zbl 1380.65021Zhang, Zhongqiang; Karniadakis, George Em 35 2017 Mechanisms of transverse motions in turbulent wall flows. Zbl 1039.76028Karniadakis, G. E.; Choi, Kwing-So 35 2003 Spectral element-Fourier methods for incompressible turbulent flows. Zbl 0722.76053Karniadakis, G. E. 34 1990 A direct numerical simulation study of flow past a freely vibrating cable. Zbl 0901.76062Newman, David J.; Karniadakis, George Em 34 1997 Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Zbl 1407.62252Perdikaris, P.; Raissi, M.; Damianou, A.; Lawrence, N. D.; Karniadakis, G. E. 33 2017 Unstructured spectral element methods for simulation of turbulent flows. Zbl 0840.76070Henderson, Ronald D.; Karniadakis, George Em 33 1995 A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. Zbl 1454.76006Meng, Xuhui; Karniadakis, George Em 32 2020 Deep learning of vortex-induced vibrations. Zbl 1415.76177Raissi, Maziar; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em 32 2019 Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. Zbl 1440.60067Zhang, Dongkun; Guo, Ling; Karniadakis, George Em 32 2020 Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. Zbl 0875.76111Kaiktsis, Lambros; Karniadakis, George Em; Orszag, Steven A. 32 1996 On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Zbl 1473.65349Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em 32 2020 Optimal error estimates of spectral Petrov-Galerkin and collocation methods for initial value problems of fractional differential equations. Zbl 1326.65100Zhang, Zhongqiang; Zeng, Fanhai; Karniadakis, George Em 31 2015 Systematic coarse-graining of spectrin-level red blood cell models. Zbl 1231.74311Fedosov, Dmitry A.; Caswell, Bruce; Karniadakis, George Em 30 2010 Equation-free/Galerkin-free POD-assisted computation of incompressible flows. Zbl 1213.76146Sirisup, Sirod; Karniadakis, George Em; Xiu, Dongbin; Kevrekidis, Ioannis G. 30 2005 Tetrahedral \(hp\) finite elements: Algorithms and flow simulations. Zbl 0847.76038Sherwin, S. J.; Karniadakis, G. E. 29 1996 Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. Zbl 0929.76095Lomtev, I.; Quillen, C. B.; Karniadakis, G. E. 29 1998 Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. Zbl 1415.74049Yu, Yue; Perdikaris, Paris; Karniadakis, George Em 28 2016 Physics-informed generative adversarial networks for stochastic differential equations. Zbl 1440.60065Yang, Liu; Zhang, Dongkun; Karniadakis, George Em 28 2020 B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. Zbl 07508507Yang, Liu; Meng, Xuhui; Karniadakis, George Em 28 2021 PPINN: parareal physics-informed neural network for time-dependent PDEs. Zbl 07337119Meng, Xuhui; Li, Zhen; Zhang, Dongkun; Karniadakis, George Em 27 2020 Drag reduction in wall-bounded turbulence via a transverse travelling wave. Zbl 1112.76373Du, Yiqing; Symeonidis, V.; Karniadakis, G. E. 27 2002 Dynamics and flow structures in the turbulent wake of rigid and flexible cylinders subject to vortex-induced vibrations. Zbl 0983.76029Evangelinos, Constantinos; Karniadakis, George Em 26 1999 Generalized polynomial chaos and random oscillators. Zbl 1060.70515Lucor, D.; Su, C.-H.; Karniadakis, G. E. 26 2004 Exact PDF equations and closure approximations for advective-reactive transport. Zbl 1349.35068Venturi, D.; Tartakovsky, D. M.; Tartakovsky, A. M.; Karniadakis, G. E. 26 2013 Spectral element simulations of laminar and turbulent flows in complex geometries. Zbl 0678.76050Karniadakis, George Em 25 1989 Stability and accuracy of periodic flow solutions obtained by a POD-penalty method. Zbl 1070.35024Sirisup, S.; Karniadakis, G. E. 25 2005 Basis functions for triangular and quadrilateral high-order elements. Zbl 0930.35016Warburton, T. C.; Sherwin, S. J.; Karniadakis, G. E. 24 1999 A convergence study of a new partitioned fluid-structure interaction algorithm based on fictitious mass and damping. Zbl 1426.76496Baek, Hyoungsu; Karniadakis, George Em 24 2012 A discontinuous Galerkin method for the Navier-Stokes equations. Zbl 0951.76041Lomtev, Igor; Karniadakis, George Em 24 1999 Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Zbl 07526130Yu, Jeremy; Lu, Lu; Meng, Xuhui; Karniadakis, George Em 9 2022 A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials. Zbl 07487692Goswami, Somdatta; Yin, Minglang; Yu, Yue; Karniadakis, George Em 7 2022 A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Zbl 07526108Lu, Lu; Meng, Xuhui; Cai, Shengze; Mao, Zhiping; Goswami, Somdatta; Zhang, Zhongqiang; Karniadakis, George Em 5 2022 Error estimates for deeponets: a deep learning framework in infinite dimensions. Zbl 07525076Lanthaler, Samuel; Mishra, Siddhartha; Karniadakis, George E. 5 2022 Physics-informed neural networks for inverse problems in supersonic flows. Zbl 07561079Jagtap, Ameya D.; Mao, Zhiping; Adams, Nikolaus; Karniadakis, George Em 5 2022 Meta-learning PINN loss functions. Zbl 07527733Psaros, Apostolos F.; Kawaguchi, Kenji; Karniadakis, George Em 4 2022 When do extended physics-informed neural networks (XPINNs) improve generalization? Zbl 1498.65183Hu, Zheyuan; Jagtap, Ameya D.; Karniadakis, George Em; Kawaguchi, Kenji 2 2022 Generative ensemble regression: Learning particle dynamics from observations of ensembles with physics-informed deep generative models. Zbl 1491.60117Yang, Liu; Daskalakis, Constantinos; Karniadakis, George E. 1 2022 Learning functional priors and posteriors from data and physics. Zbl 07523820Meng, Xuhui; Yang, Liu; Mao, Zhiping; del Águila Ferrandis, José; Karniadakis, George Em 1 2022 DeepXDE: a deep learning library for solving differential equations. Zbl 1459.65002Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em 87 2021 hp-VPINNs: variational physics-informed neural networks with domain decomposition. Zbl 07338008Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M. 39 2021 B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. Zbl 07508507Yang, Liu; Meng, Xuhui; Karniadakis, George Em 28 2021 NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations. Zbl 07510065Jin, Xiaowei; Cai, Shengze; Li, Hui; Karniadakis, George Em 12 2021 Non-invasive inference of thrombus material properties with physics-informed neural networks. Zbl 07340448Yin, Minglang; Zheng, Xiaoning; Humphrey, Jay D.; Karniadakis, George Em 11 2021 Two-point stress-strain-rate correlation structure and non-local eddy viscosity in turbulent flows. Zbl 1461.76181Clark Di Leoni, Patricio; Zaki, Tamer A.; Karniadakis, George; Meneveau, Charles 10 2021 Towards a unified theory of fractional and nonlocal vector calculus. Zbl 1498.26008D’Elia, Marta; Gulian, Mamikon; Olson, Hayley; Karniadakis, George Em 8 2021 Parallel physics-informed neural networks via domain decomposition. Zbl 07516435Shukla, Khemraj; Jagtap, Ameya D.; Karniadakis, George Em 8 2021 Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks. Zbl 1480.35377Chen, Xiaoli; Yang, Liu; Duan, Jinqiao; Karniadakis, George Em 7 2021 Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation. Zbl 07516434Lou, Qin; Meng, Xuhui; Karniadakis, George Em 7 2021 Multi-fidelity Bayesian neural networks: algorithms and applications. Zbl 07505956Meng, Xuhui; Babaee, Hessam; Karniadakis, George Em 6 2021 Learning and meta-learning of stochastic advection-diffusion-reaction systems from sparse measurements. Zbl 07441294Chen, Xiaoli; Duan, Jinqiao; Karniadakis, George Em 4 2021 DeepM&Mnet for hypersonics: predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators. Zbl 07516442Mao, Zhiping; Lu, Lu; Marxen, Olaf; Zaki, Tamer A.; Karniadakis, George Em 4 2021 Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. Zbl 1461.76417Cai, Shengze; Wang, Zhicheng; Fuest, Frederik; Jeon, Young Jin; Gray, Callum; Karniadakis, George Em 3 2021 DeepM&Mnet: inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Zbl 07513856Cai, Shengze; Wang, Zhicheng; Lu, Lu; Zaki, Tamer A.; Karniadakis, George Em 3 2021 A seamless multiscale operator neural network for inferring bubble dynamics. Zbl 1495.76119Lin, Chensen; Maxey, Martin; Li, Zhen; Karniadakis, George Em 2 2021 A fast multi-fidelity method with uncertainty quantification for complex data correlations: application to vortex-induced vibrations of marine risers. Zbl 07427366Meng, Xuhui; Wang, Zhicheng; Fan, Dixia; Triantafyllou, Michael S.; Karniadakis, George Em 1 2021 Multiscale parareal algorithm for long-time mesoscopic simulations of microvascular blood flow in zebrafish. Zbl 1479.76123Blumers, Ansel L.; Yin, Minglang; Nakajima, Hiroyuki; Hasegawa, Yosuke; Li, Zhen; Karniadakis, George Em 1 2021 A phase-field method for boiling heat transfer. Zbl 07503725Wang, Zhicheng; Zheng, Xiaoning; Chryssostomidis, Chryssostomos; Karniadakis, George Em 1 2021 Active- and transfer-learning applied to microscale-macroscale coupling to simulate viscoelastic flows. Zbl 07510254Zhao, Lifei; Li, Zhen; Wang, Zhicheng; Caswell, Bruce; Ouyang, Jie; Karniadakis, George Em 1 2021 What is the fractional Laplacian? A comparative review with new results. Zbl 1453.35179Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaoning; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em 72 2020 Physics-informed neural networks for high-speed flows. Zbl 1442.76092Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em 58 2020 Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Zbl 1442.92002Jagtap, Ameya D.; Kharazmi, Ehsan; Karniadakis, George Em 50 2020 Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Zbl 1478.76057Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em. 49 2020 Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Zbl 1453.68165Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em 41 2020 Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Zbl 07419158Jagtap, Ameya D.; Karniadakis, George Em 41 2020 A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. Zbl 1454.76006Meng, Xuhui; Karniadakis, George Em 32 2020 Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. Zbl 1440.60067Zhang, Dongkun; Guo, Ling; Karniadakis, George Em 32 2020 On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Zbl 1473.65349Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em 32 2020 Physics-informed generative adversarial networks for stochastic differential equations. Zbl 1440.60065Yang, Liu; Zhang, Dongkun; Karniadakis, George Em 28 2020 PPINN: parareal physics-informed neural network for time-dependent PDEs. Zbl 07337119Meng, Xuhui; Li, Zhen; Zhang, Dongkun; Karniadakis, George Em 27 2020 Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Zbl 1472.68175Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em 19 2020 nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications. Zbl 07508384Pang, G.; D’Elia, M.; Parks, M.; Karniadakis, G. E. 11 2020 Sympnets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Zbl 1475.68316Jin, Pengzhan; Zhang, Zhen; Zhu, Aiqing; Tang, Yifa; Karniadakis, George Em 11 2020 A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations. Zbl 1453.65370Zhang, Hui; Jiang, Xiaoyun; Zeng, Fanhai; Karniadakis, George Em 9 2020 Dying ReLU and initialization: theory and numerical examples. Zbl 07419148Lu, Lu; Shin, Yeonjong; Su, Yanhui; Karniadakis, George Em 6 2020 A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation. Zbl 1442.65144Li, Xianjuan; Mao, Zhiping; Wang, Nan; Song, Fangying; Wang, Hong; Karniadakis, George Em 3 2020 Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness. Zbl 1475.68315Jin, Pengzhan; Lu, Lu; Tang, Yifa; Karniadakis, George Em 2 2020 Physics-informed semantic inpainting: application to geostatistical modeling. Zbl 07507237Zheng, Qiang; Zeng, Lingzao; Karniadakis, George Em 1 2020 Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Zbl 1415.68175Raissi, M.; Perdikaris, P.; Karniadakis, G. E. 491 2019 FPINNs: fractional physics-informed neural networks. Zbl 1420.35459Pang, Guofei; Lu, Lu; Karniadakis, George Em 68 2019 Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Zbl 1454.65008Zhang, Dongkun; Lu, Lu; Guo, Ling; Karniadakis, George Em 40 2019 Deep learning of vortex-induced vibrations. Zbl 1415.76177Raissi, Maziar; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em 32 2019 Efficient multistep methods for tempered fractional calculus: algorithms and simulations. Zbl 07099350Guo, Ling; Zeng, Fanhai; Turner, Ian; Burrage, Kevin; Karniadakis, George Em 14 2019 Machine learning of space-fractional differential equations. Zbl 1419.35209Gulian, Mamikon; Raissi, Maziar; Perdikaris, Paris; Karniadakis, George 10 2019 Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations. Zbl 1440.65258Zhao, Tinggang; Mao, Zhiping; Karniadakis, George Em 9 2019 Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network. Zbl 1434.76053Mehta, Pavan Pranjivan; Pang, Guofei; Song, Fangying; Karniadakis, George Em 9 2019 Neural-net-induced Gaussian process regression for function approximation and PDE solution. Zbl 1451.68242Pang, Guofei; Yang, Liu; Karniadakis, George Em 7 2019 Fractional Gray-Scott model: well-posedness, discretization, and simulations. Zbl 1440.35344Wang, Tingting; Song, Fangying; Wang, Hong; Karniadakis, George Em 6 2019 Fractional magneto-hydrodynamics: algorithms and applications. Zbl 1416.76127Song, Fangying; Karniadakis, George Em 4 2019 An entropy-viscosity large eddy simulation study of turbulent flow in a flexible pipe. Zbl 1415.76369Wang, Zhicheng; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em 4 2019 Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations. Zbl 1449.35447Mao, Zhiping; Li, Zhen; Karniadakis, George Em 4 2019 Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow. Zbl 1430.76091Fan, Dixia; Wang, Zhicheng; Triantafyllou, Michael S.; Karniadakis, George Em 3 2019 Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: application to hydrodynamics. Zbl 1452.76172Blumers, Ansel L.; Li, Zhen; Karniadakis, George Em 2 2019 A stabilized phase-field method for two-phase flow at high Reynolds number and large density/viscosity ratio. Zbl 1453.76154Wang, Zhicheng; Dong, Suchuan; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em 2 2019 A spectral penalty method for two-sided fractional differential equations with general boundary conditions. Zbl 1501.65143Wang, Nan; Mao, Zhiping; Huang, Chengming; Karniadakis, George Em 2 2019 One-dimensional modeling of fractional flow reserve in coronary artery disease: uncertainty quantification and Bayesian optimization. Zbl 1441.76150Yin, Minglang; Yazdani, Alireza; Karniadakis, George Em 2 2019 A Computational Mechanics special issue on: Data-driven modeling and simulation – theory, methods, and applications. Zbl 1476.00073 1 2019 Numerical methods. Zbl 1410.65001 1 2019 Hidden physics models: machine learning of nonlinear partial differential equations. Zbl 1381.68248Raissi, Maziar; Karniadakis, George Em 143 2018 Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. Zbl 1386.65030Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 64 2018 A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. Zbl 1422.65428Mao, Zhiping; Karniadakis, George Em 47 2018 A Riesz basis Galerkin method for the tempered fractional Laplacian. Zbl 1402.65120Zhang, Zhijiang; Deng, Weihua; Karniadakis, George Em 18 2018 A partitioned coupling framework for peridynamics and classical theory: analysis and simulations. Zbl 1440.74045Yu, Yue; Bargos, Fabiano F.; You, Huaiqian; Parks, Michael L.; Bittencourt, Marco L.; Karniadakis, George E. 14 2018 A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations. Zbl 1404.65105Zeng, Fanhai; Turner, Ian; Burrage, Kevin; Karniadakis, George Em 13 2018 Multi-fidelity optimization of super-cavitating hydrofoils. Zbl 1439.74256Bonfiglio, L.; Perdikaris, P.; Brizzolara, S.; Karniadakis, G. E. 7 2018 A dissipative particle dynamics method for arbitrarily complex geometries. Zbl 1380.76123Li, Zhen; Bian, Xin; Tang, Yu-Hang; Karniadakis, George Em 6 2018 A computational stochastic methodology for the design of random meta-materials under geometric constraints. Zbl 1398.60073Tsantili, Ivi C.; Cho, Min Hyung; Cai, Wei; Karniadakis, George Em 6 2018 Stochastic domain decomposition via moment minimization. Zbl 1391.60167Zhang, Dongkun; Babaee, Hessam; Karniadakis, George Em 3 2018 A spectral-element/Fourier smoothed profile method for large-eddy simulations of complex VIV problems. Zbl 1410.76320Wang, Zhicheng; Triantafyllou, Michael S.; Constantinides, Yiannis; Karniadakis, George Em 3 2018 Active learning of constitutive relation from mesoscopic dynamics for macroscopic modeling of non-Newtonian flows. Zbl 1392.76058Zhao, Lifei; Li, Zhen; Caswell, Bruce; Ouyang, Jie; Karniadakis, George Em 2 2018 Analytical and computational studies of correlations of hydrodynamic fluctuations in shear flow. Zbl 1488.76101Bian, Xin; Deng, Mingge; Karniadakis, George Em 1 2018 Machine learning of linear differential equations using Gaussian processes. Zbl 1380.68339Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 97 2017 Inferring solutions of differential equations using noisy multi-fidelity data. Zbl 1382.65229Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em 95 2017 A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. Zbl 1431.65193Zeng, Fanhai; Mao, Zhiping; Karniadakis, George Em 40 2017 Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Zbl 1439.65081Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em 38 2017 Petrov-Galerkin and spectral collocation methods for distributed order differential equations. Zbl 1367.65113Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em 38 2017 Numerical methods for stochastic partial differential equations with white noise. Zbl 1380.65021Zhang, Zhongqiang; Karniadakis, George Em 35 2017 Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Zbl 1407.62252Perdikaris, P.; Raissi, M.; Damianou, A.; Lawrence, N. D.; Karniadakis, G. E. 33 2017 Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. Zbl 1380.65357Song, Fangying; Xu, Chuanju; Karniadakis, George Em 23 2017 A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. Zbl 1367.65108Lischke, Anna; Zayernouri, Mohsen; Karniadakis, George Em 23 2017 Multi-fidelity Gaussian process regression for prediction of random fields. Zbl 1419.62272Parussini, L.; Venturi, D.; Perdikaris, P.; Karniadakis, G. E. 19 2017 Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization. Zbl 1419.62499Pang, Guofei; Perdikaris, Paris; Cai, Wei; Karniadakis, George Em 19 2017 Adaptive finite element method for fractional differential equations using hierarchical matrices. Zbl 1439.65091Zhao, Xuan; Hu, Xiaozhe; Cai, Wei; Karniadakis, George Em 15 2017 A Petrov-Galerkin spectral element method for fractional elliptic problems. Zbl 1439.65205Kharazmi, Ehsan; Zayernouri, Mohsen; Karniadakis, George Em 13 2017 A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems. Zbl 1380.76093Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em 10 2017 A tunable finite difference method for fractional differential equations with non-smooth solutions. Zbl 1439.65082Chen, Xuejuan; Zeng, Fanhai; Karniadakis, George Em 9 2017 Fractional Burgers equation with nonlinear non-locality: spectral vanishing viscosity and local discontinuous Galerkin methods. Zbl 1380.65280Mao, Zhiping; Karniadakis, George Em 8 2017 Dissipative particle dynamics: foundation, evolution, implementation, and applications. Zbl 1387.35496Li, Z.; Bian, X.; Li, X.; Deng, M.; Tang, Y.-H.; Caswell, B.; Karniadakis, George E. 4 2017 Fractional spectral vanishing viscosity method: application to the quasi-geostrophic equation. Zbl 1422.65289Song, Fangying; Karniadakis, George Em 1 2017 Numerical methods for high-dimensional kinetic equations. Zbl 1404.65117Cho, Heyrim; Venturi, Daniele; Karniadakis, George Em 1 2017 ...and 202 more Documents all cited Publications top 5 cited Publications all top 5 Cited by 8,163 Authors 197 Karniadakis, George Em 72 Sherwin, Spencer J. 54 Lin, Guang 51 Wang, Hong 46 Kirby, Robert M. II 40 Xiu, Dongbin 39 Zheng, Xiangcheng 31 Thompson, Mark Christopher 30 Doostan, Alireza 30 Narayan, Akil C. 26 Shen, Jie 25 Dehghan Takht Fooladi, Mehdi 25 Le Maître, Olivier P. 24 Ghanem, Roger G. 24 Perdikaris, Paris G. 24 Schwab, Christoph 24 Zaky, Mahmoud A. 23 Hesthaven, Jan S. 23 Rozza, Gianluigi 23 Shu, Chi-Wang 23 Venturi, Daniele 23 Zhou, Tao 21 Hourigan, Kerry 21 Kopriva, David Alan 21 Zabaras, Nicholas J. 20 Blackburn, Hugh M. 20 Dumbser, Michael 20 Nobile, Fabio 20 Tartakovsky, Daniel M. 20 Zayernouri, Mohsen 19 Mao, Zhiping 18 Dong, Suchuan 18 Jin, Shi 18 Knio, Omar M. 18 Lucor, Didier 18 Quarteroni, Alfio M. 18 Wan, Xiaoliang 18 Wang, Lilian 18 Xu, Chuanju 17 Abbaszadeh, Mostafa 17 Liu, Fawang 17 Sagaut, Pierre 17 Yu, Yue 17 Zhang, Dongxiao 16 Brunton, Steven L. 16 Guo, Ben-Yu 16 Guo, Ling 16 Jakeman, John Davis 16 Li, Hong 16 Li, Zhen 16 Najm, Habib N. 16 Noack, Bernd R. 16 Nordström, Jan 16 Yang, Xiu 16 Zeng, Fanhai 15 Ainsworth, Mark 15 Cheng, Liang 15 Elman, Howard C. 15 Giraldo, Francis X. 15 Iaccarino, Gianluca 15 Jiang, Xiaoyun 15 Liu, Yang 15 Mishra, Siddhartha 15 Tartakovsky, Alexandre M. 15 Warburton, Timothy 15 Xing, Yulong 15 Zhang, Zhimin 15 Zhang, Zhongqiang 14 Baccouch, Mahboub 14 Boscheri, Walter 14 Bourguet, Rémi 14 D’Elia, Marta 14 Ferrer, Esteban 14 Kronbichler, Martin 14 Machado, José António Tenreiro 14 Pasquetti, Richard 14 Sheard, Gregory John 14 Soize, Christian 14 Turner, Ian William 13 Băleanu, Dumitru I. 13 Cortés López, Juan Carlos 13 Deng, Weihua 13 Gassner, Gregor J. 13 Gerritsma, Marc I. 13 Kevrekidis, Ioannis George 13 Kutz, J. 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