Anagnostopoulos, Sokratis J.; Toscano, Juan Diego; Stergiopulos, Nikolaos; Karniadakis, George Em Residual-based attention in physics-informed neural networks. (English) Zbl 07824894 Comput. Methods Appl. Mech. Eng. 421, Article ID 116805, 19 p. (2024). MSC: 92-XX 68-XX PDFBibTeX XMLCite \textit{S. J. Anagnostopoulos} et al., Comput. Methods Appl. Mech. Eng. 421, Article ID 116805, 19 p. (2024; Zbl 07824894) Full Text: DOI
Haghighat, Ehsan; Waheed, Umair bin; Karniadakis, George En-DeepONet: an enrichment approach for enhancing the expressivity of neural operators with applications to seismology. (English) Zbl 07820233 Comput. Methods Appl. Mech. Eng. 420, Article ID 116681, 14 p. (2024). MSC: 68-XX 81-XX PDFBibTeX XMLCite \textit{E. Haghighat} et al., Comput. Methods Appl. Mech. Eng. 420, Article ID 116681, 14 p. (2024; Zbl 07820233) Full Text: DOI arXiv
Zou, Zongren; Meng, Xuhui; Psaros, Apostolos F.; Karniadakis, George E. NeuralUQ: a comprehensive library for uncertainty quantification in neural differential equations and operators. (English) Zbl 07805456 SIAM Rev. 66, No. 1, 161-190 (2024). MSC: 68T07 PDFBibTeX XMLCite \textit{Z. Zou} et al., SIAM Rev. 66, No. 1, 161--190 (2024; Zbl 07805456) Full Text: DOI arXiv
Goswami, Somdatta; Jagtap, Ameya D.; Babaee, Hessam; Susi, Bryan T.; Karniadakis, George Em Learning stiff chemical kinetics using extended deep neural operators. (English) Zbl 07796694 Comput. Methods Appl. Mech. Eng. 419, Article ID 116674, 20 p. (2024). MSC: 92-XX 80-XX PDFBibTeX XMLCite \textit{S. Goswami} et al., Comput. Methods Appl. Mech. Eng. 419, Article ID 116674, 20 p. (2024; Zbl 07796694) Full Text: DOI arXiv
Zhang, Zhen; Zou, Zongren; Kuhl, Ellen; Karniadakis, George Em Discovering a reaction-diffusion model for Alzheimer’s disease by combining PINNs with symbolic regression. (English) Zbl 07796670 Comput. Methods Appl. Mech. Eng. 419, Article ID 116647, 14 p. (2024). MSC: 92-XX 35-XX PDFBibTeX XMLCite \textit{Z. Zhang} et al., Comput. Methods Appl. Mech. Eng. 419, Article ID 116647, 14 p. (2024; Zbl 07796670) Full Text: DOI arXiv
Zhuang, Qiao; Yao, Chris Ziyi; Zhang, Zhongqiang; Karniadakis, George Em Two-scale Neural Networks for Partial Differential Equations with Small Parameters. arXiv:2402.17232 Preprint, arXiv:2402.17232 [math.NA] (2024). MSC: 65N35 35B25 BibTeX Cite \textit{Q. Zhuang} et al., ``Two-scale Neural Networks for Partial Differential Equations with Small Parameters'', Preprint, arXiv:2402.17232 [math.NA] (2024) Full Text: arXiv OA License
Hu, Zheyuan; Zhang, Zhongqiang; Karniadakis, George Em; Kawaguchi, Kenji Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations. arXiv:2402.07465 Preprint, arXiv:2402.07465 [cs.LG] (2024). MSC: 14J60 BibTeX Cite \textit{Z. Hu} et al., ``Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations'', Preprint, arXiv:2402.07465 [cs.LG] (2024) Full Text: arXiv OA License
Kopaničáková, Alena; Karniadakis, George Em DeepOnet Based Preconditioning Strategies For Solving Parametric Linear Systems of Equations. arXiv:2401.02016 Preprint, arXiv:2401.02016 [math.NA] (2024). BibTeX Cite \textit{A. Kopaničáková} and \textit{G. E. Karniadakis}, ``DeepOnet Based Preconditioning Strategies For Solving Parametric Linear Systems of Equations'', Preprint, arXiv:2401.02016 [math.NA] (2024) Full Text: arXiv OA License
Ma, Lei; Li, Rongxin; Zeng, Fanhai; Guo, Ling; Karniadakis, George Em Bi-orthogonal fPINN: a physics-informed neural network method for solving time-dependent stochastic fractional PDEs. (English) Zbl 07783920 Commun. Comput. Phys. 34, No. 4, 1133-1176 (2023). MSC: 65M32 68T07 68Q32 65C05 65M06 65D32 49M41 65M12 65M15 35R30 35R60 26A33 35R11 PDFBibTeX XMLCite \textit{L. Ma} et al., Commun. Comput. Phys. 34, No. 4, 1133--1176 (2023; Zbl 07783920) Full Text: DOI arXiv
Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em Accelerating gradient descent and Adam via fractional gradients. (English) Zbl 07755615 Neural Netw. 161, 185-201 (2023). Reviewer: Haydar Akca (Abu Dhabi) MSC: 90C26 90C20 PDFBibTeX XMLCite \textit{Y. Shin} et al., Neural Netw. 161, 185--201 (2023; Zbl 07755615) Full Text: DOI
Penwarden, Michael; Jagtap, Ameya D.; Zhe, Shandian; Karniadakis, George Em; Kirby, Robert M. A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions. (English) Zbl 07748050 J. Comput. Phys. 493, Article ID 112464, 21 p. (2023). MSC: 65Mxx 68Txx 65-XX PDFBibTeX XMLCite \textit{M. Penwarden} et al., J. Comput. Phys. 493, Article ID 112464, 21 p. (2023; Zbl 07748050) Full Text: DOI arXiv
Howard, Amanda A.; Perego, Mauro; Karniadakis, George Em; Stinis, Panos Multifidelity deep operator networks for data-driven and physics-informed problems. (English) Zbl 07748048 J. Comput. Phys. 493, Article ID 112462, 28 p. (2023). MSC: 68Txx 65Mxx 65Cxx PDFBibTeX XMLCite \textit{A. A. Howard} et al., J. Comput. Phys. 493, Article ID 112462, 28 p. (2023; Zbl 07748048) Full Text: DOI
He, QiZhi; Perego, Mauro; Howard, Amanda A.; Karniadakis, George Em; Stinis, Panos A hybrid deep neural operator/finite element method for ice-sheet modeling. (English) Zbl 07742911 J. Comput. Phys. 492, Article ID 112428, 17 p. (2023). MSC: 86Axx 76Mxx 65Cxx PDFBibTeX XMLCite \textit{Q. He} et al., J. Comput. Phys. 492, Article ID 112428, 17 p. (2023; Zbl 07742911) Full Text: DOI arXiv
Kiyani, Elham; Shukla, Khemraj; Karniadakis, George Em; Karttunen, Mikko A framework based on symbolic regression coupled with eXtended physics-informed neural networks for gray-box learning of equations of motion from data. (English) Zbl 07736350 Comput. Methods Appl. Mech. Eng. 415, Article ID 116258, 21 p. (2023). MSC: 68-XX 86-XX PDFBibTeX XMLCite \textit{E. Kiyani} et al., Comput. Methods Appl. Mech. Eng. 415, Article ID 116258, 21 p. (2023; Zbl 07736350) Full Text: DOI arXiv
Kahana, Adar; Zhang, Enrui; Goswami, Somdatta; Karniadakis, George; Ranade, Rishikesh; Pathak, Jay On the geometry transferability of the hybrid iterative numerical solver for differential equations. (English) Zbl 07719582 Comput. Mech. 72, No. 3, 471-484 (2023). MSC: 74S99 76M99 74B05 76S05 68T07 68T05 PDFBibTeX XMLCite \textit{A. Kahana} et al., Comput. Mech. 72, No. 3, 471--484 (2023; Zbl 07719582) Full Text: DOI arXiv
Peyvan, Ahmad; Shukla, Khemraj; Chan, Jesse; Karniadakis, George High-order methods for hypersonic flows with strong shocks and real chemistry. (English) Zbl 07715240 J. Comput. Phys. 490, Article ID 112310, 25 p. (2023). MSC: 65Mxx 76Mxx 35Lxx PDFBibTeX XMLCite \textit{A. Peyvan} et al., J. Comput. Phys. 490, Article ID 112310, 25 p. (2023; Zbl 07715240) Full Text: DOI arXiv
Li, Z. (ed.); Hu, Guohui (ed.); Wang, Zhiliang (ed.); Karniadakis, G. E. (ed.) Preface: machine-learning approaches for computational mechanics. (English) Zbl 07712695 AMM, Appl. Math. Mech., Engl. Ed. 44, No. 7, 1035-1038 (2023). MSC: 00Bxx PDFBibTeX XMLCite \textit{Z. Li} (ed.) et al., AMM, Appl. Math. Mech., Engl. Ed. 44, No. 7, 1035--1038 (2023; Zbl 07712695) Full Text: DOI
Zhu, Min; Zhang, Handi; Jiao, Anran; Karniadakis, George Em; Lu, Lu Reliable extrapolation of deep neural operators informed by physics or sparse observations. (English) Zbl 07692973 Comput. Methods Appl. Mech. Eng. 412, Article ID 116064, 36 p. (2023). MSC: 68-XX 92-XX PDFBibTeX XMLCite \textit{M. Zhu} et al., Comput. Methods Appl. Mech. Eng. 412, Article ID 116064, 36 p. (2023; Zbl 07692973) Full Text: DOI arXiv
Kontolati, Katiana; Goswami, Somdatta; Shields, Michael D.; Karniadakis, George Em On the influence of over-parameterization in manifold based surrogates and deep neural operators. (English) Zbl 07660362 J. Comput. Phys. 479, Article ID 112008, 18 p. (2023). MSC: 65Cxx 68Txx 60Hxx PDFBibTeX XMLCite \textit{K. Kontolati} et al., J. Comput. Phys. 479, Article ID 112008, 18 p. (2023; Zbl 07660362) Full Text: DOI arXiv
Psaros, Apostolos F.; Meng, Xuhui; Zou, Zongren; Guo, Ling; Karniadakis, George Em Uncertainty quantification in scientific machine learning: methods, metrics, and comparisons. (English) Zbl 07652802 J. Comput. Phys. 477, Article ID 111902, 83 p. (2023). MSC: 68Txx 65Cxx 62Fxx PDFBibTeX XMLCite \textit{A. F. Psaros} et al., J. Comput. Phys. 477, Article ID 111902, 83 p. (2023; Zbl 07652802) Full Text: DOI arXiv
Clark Di Leoni, Patricio; Lu, Lu; Meneveau, Charles; Karniadakis, George Em; Zaki, Tamer A. Neural operator prediction of linear instability waves in high-speed boundary layers. (English) Zbl 07640550 J. Comput. Phys. 474, Article ID 111793, 22 p. (2023). MSC: 76-XX 86-XX PDFBibTeX XMLCite \textit{P. Clark Di Leoni} et al., J. Comput. Phys. 474, Article ID 111793, 22 p. (2023; Zbl 07640550) Full Text: DOI arXiv
De Florio, Mario; Kahana, Adar; Karniadakis, George Em Analysis of biologically plausible neuron models for regression with spiking neural networks. arXiv:2401.00369 Preprint, arXiv:2401.00369 [math.NA] (2023). BibTeX Cite \textit{M. De Florio} et al., ``Analysis of biologically plausible neuron models for regression with spiking neural networks'', Preprint, arXiv:2401.00369 [math.NA] (2023) Full Text: arXiv OA License
Hu, Zheyuan; Shi, Zekun; Karniadakis, George Em; Kawaguchi, Kenji Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks. arXiv:2312.14499 Preprint, arXiv:2312.14499 [cs.LG] (2023). MSC: 14J60 BibTeX Cite \textit{Z. Hu} et al., ``Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks'', Preprint, arXiv:2312.14499 [cs.LG] (2023) Full Text: DOI arXiv OA License
Hu, Zheyuan; Yang, Zhouhao; Wang, Yezhen; Karniadakis, George Em; Kawaguchi, Kenji Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs. arXiv:2311.15283 Preprint, arXiv:2311.15283 [cs.LG] (2023). MSC: 14J60 BibTeX Cite \textit{Z. Hu} et al., ``Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs'', Preprint, arXiv:2311.15283 [cs.LG] (2023) Full Text: arXiv OA License
Chen, Paula; Meng, Tingwei; Zou, Zongren; Darbon, Jérôme; Karniadakis, George Em Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning. arXiv:2311.07790 Preprint, arXiv:2311.07790 [cs.LG] (2023). BibTeX Cite \textit{P. Chen} et al., ``Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning'', Preprint, arXiv:2311.07790 [cs.LG] (2023) Full Text: arXiv OA License
Lin, Bin; Mao, Zhiping; Wang, Zhicheng; Karniadakis, George Em Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions. arXiv:2310.19590 Preprint, arXiv:2310.19590 [cs.LG] (2023). BibTeX Cite \textit{B. Lin} et al., ``Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions'', Preprint, arXiv:2310.19590 [cs.LG] (2023) Full Text: arXiv OA License
Hu, Zheyuan; Shukla, Khemraj; Karniadakis, George Em; Kawaguchi, Kenji Tackling the Curse of Dimensionality with Physics-Informed Neural Networks. arXiv:2307.12306 Preprint, arXiv:2307.12306 [cs.LG] (2023). MSC: 14J60 BibTeX Cite \textit{Z. Hu} et al., ``Tackling the Curse of Dimensionality with Physics-Informed Neural Networks'', Preprint, arXiv:2307.12306 [cs.LG] (2023) Full Text: arXiv OA License
Ovadia, Oded; Oommen, Vivek; Kahana, Adar; Peyvan, Ahmad; Turkel, Eli; Karniadakis, George Em Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO). arXiv:2307.09072 Preprint, arXiv:2307.09072 [cs.LG] (2023). BibTeX Cite \textit{O. Ovadia} et al., ``Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)'', Preprint, arXiv:2307.09072 [cs.LG] (2023) Full Text: arXiv OA License
Varghese, Alan John; Bora, Aniruddha; Xu, Mengjia; Karniadakis, George Em TransformerG2G: Adaptive time-stepping for learning temporal graph embeddings using transformers. arXiv:2307.02588 Preprint, arXiv:2307.02588 [cs.LG] (2023). BibTeX Cite \textit{A. J. Varghese} et al., ``TransformerG2G: Adaptive time-stepping for learning temporal graph embeddings using transformers'', Preprint, arXiv:2307.02588 [cs.LG] (2023) Full Text: arXiv OA License
Kopaničáková, Alena; Kothari, Hardik; Karniadakis, George Em; Krause, Rolf Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies. arXiv:2306.17648 Preprint, arXiv:2306.17648 [math.NA] (2023). MSC: 90C30 90C26 90C06 65M55 68T07 BibTeX Cite \textit{A. Kopaničáková} et al., ``Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies'', Preprint, arXiv:2306.17648 [math.NA] (2023) Full Text: arXiv OA License
Shekarpaz, Simin; Zeng, Fanhai; Karniadakis, George Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models. arXiv:2304.13205 Preprint, arXiv:2304.13205 [math.NA] (2023). BibTeX Cite \textit{S. Shekarpaz} et al., ``Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models'', Preprint, arXiv:2304.13205 [math.NA] (2023) Full Text: arXiv OA License
Kontolati, Katiana; Goswami, Somdatta; Karniadakis, George Em; Shields, Michael D. Learning in latent spaces improves the predictive accuracy of deep neural operators. arXiv:2304.07599 Preprint, arXiv:2304.07599 [cs.LG] (2023). BibTeX Cite \textit{K. Kontolati} et al., ``Learning in latent spaces improves the predictive accuracy of deep neural operators'', Preprint, arXiv:2304.07599 [cs.LG] (2023) Full Text: arXiv OA License
Chen, Paula; Meng, Tingwei; Zou, Zongren; Darbon, Jérôme; Karniadakis, George Em Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems. arXiv:2303.12928 Preprint, arXiv:2303.12928 [cs.LG] (2023). MSC: 35F21 49N05 49N10 68T05 35B37 BibTeX Cite \textit{P. Chen} et al., ``Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems'', Preprint, arXiv:2303.12928 [cs.LG] (2023) Full Text: arXiv OA License
Ovadia, Oded; Kahana, Adar; Stinis, Panos; Turkel, Eli; Karniadakis, George Em ViTO: Vision Transformer-Operator. arXiv:2303.08891 Preprint, arXiv:2303.08891 [cs.CV] (2023). BibTeX Cite \textit{O. Ovadia} et al., ``ViTO: Vision Transformer-Operator'', Preprint, arXiv:2303.08891 [cs.CV] (2023) Full Text: arXiv OA License
Shukla, Khemraj; Oommen, Vivek; Peyvan, Ahmad; Penwarden, Michael; Bravo, Luis; Ghoshal, Anindya; Kirby, Robert M.; Karniadakis, George Em Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils. arXiv:2302.00807 Preprint, arXiv:2302.00807 [physics.flu-dyn] (2023). BibTeX Cite \textit{K. Shukla} et al., ``Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils'', Preprint, arXiv:2302.00807 [physics.flu-dyn] (2023) Full Text: arXiv OA License
Chen, Xuejuan; Mao, Zhiping; Karniadakis, George Em Efficient and accurate numerical methods using the accelerated spectral deferred correction for solving fractional differential equations. (English) Zbl 1524.65643 Numer. Math., Theory Methods Appl. 15, No. 4, 876-902 (2022). MSC: 65M70 65N35 65E05 41A05 41A10 41A25 26A33 35M11 65F10 65F08 34A08 65N50 65R20 PDFBibTeX XMLCite \textit{X. Chen} et al., Numer. Math., Theory Methods Appl. 15, No. 4, 876--902 (2022; Zbl 1524.65643) Full Text: DOI
Linka, Kevin; Schäfer, Amelie; Meng, Xuhui; Zou, Zongren; Karniadakis, George Em; Kuhl, Ellen Bayesian physics informed neural networks for real-world nonlinear dynamical systems. (English) Zbl 1507.68246 Comput. Methods Appl. Mech. Eng. 402, Article ID 115346, 22 p. (2022). MSC: 68T05 37M10 62F15 PDFBibTeX XMLCite \textit{K. Linka} et al., Comput. Methods Appl. Mech. Eng. 402, Article ID 115346, 22 p. (2022; Zbl 1507.68246) Full Text: DOI arXiv
Yin, Minglang; Zhang, Enrui; Yu, Yue; Karniadakis, George Em Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems. (English) Zbl 1507.74523 Comput. Methods Appl. Mech. Eng. 402, Article ID 115027, 26 p. (2022). MSC: 74S05 68T07 PDFBibTeX XMLCite \textit{M. Yin} et al., Comput. Methods Appl. Mech. Eng. 402, Article ID 115027, 26 p. (2022; Zbl 1507.74523) Full Text: DOI arXiv
Zhang, Hui; Zeng, Fanhai; Jiang, Xiaoyun; Karniadakis, George Em Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations. (English) Zbl 1503.65194 Fract. Calc. Appl. Anal. 25, No. 2, 453-487 (2022). MSC: 65M06 35R11 65M15 65M12 26A33 PDFBibTeX XMLCite \textit{H. Zhang} et al., Fract. Calc. Appl. Anal. 25, No. 2, 453--487 (2022; Zbl 1503.65194) Full Text: DOI arXiv
Meng, Tingwei; Zhang, Zhen; Darbon, Jerome; Karniadakis, George SympOCnet: solving optimal control problems with applications to high-dimensional multiagent path planning problems. (English) Zbl 1504.49052 SIAM J. Sci. Comput. 44, No. 6, B1341-B1368 (2022). MSC: 49M37 49M20 49M29 68T07 PDFBibTeX XMLCite \textit{T. Meng} et al., SIAM J. Sci. Comput. 44, No. 6, B1341--B1368 (2022; Zbl 1504.49052) Full Text: DOI arXiv
Hu, Zheyuan; Jagtap, Ameya D.; Karniadakis, George Em; Kawaguchi, Kenji When do extended physics-informed neural networks (XPINNs) improve generalization? (English) Zbl 1498.65183 SIAM J. Sci. Comput. 44, No. 5, A3158-A3182 (2022). MSC: 65M99 68Q32 68T05 65M55 PDFBibTeX XMLCite \textit{Z. Hu} et al., SIAM J. Sci. Comput. 44, No. 5, A3158--A3182 (2022; Zbl 1498.65183) Full Text: DOI arXiv
Jagtap, Ameya D.; Mao, Zhiping; Adams, Nikolaus; Karniadakis, George Em Physics-informed neural networks for inverse problems in supersonic flows. (English) Zbl 07561079 J. Comput. Phys. 466, Article ID 111402, 18 p. (2022). MSC: 68Txx 35Lxx 76Mxx PDFBibTeX XMLCite \textit{A. D. Jagtap} et al., J. Comput. Phys. 466, Article ID 111402, 18 p. (2022; Zbl 07561079) Full Text: DOI arXiv
Deng, Mingge; Tushar, Faisal; Bravo, Luis; Ghoshal, Anindya; Karniadakis, George; Li, Zhen Theory and simulation of electrokinetic fluctuations in electrolyte solutions at the mesoscale. (English) Zbl 1512.76133 J. Fluid Mech. 942, Paper No. A29, 22 p. (2022). MSC: 76W05 76T20 76M35 PDFBibTeX XMLCite \textit{M. Deng} et al., J. Fluid Mech. 942, Paper No. A29, 22 p. (2022; Zbl 1512.76133) Full Text: DOI arXiv
Psaros, Apostolos F.; Kawaguchi, Kenji; Karniadakis, George Em Meta-learning PINN loss functions. (English) Zbl 07527733 J. Comput. Phys. 458, Article ID 111121, 37 p. (2022). MSC: 68Txx 65-XX 65Mxx PDFBibTeX XMLCite \textit{A. F. Psaros} et al., J. Comput. Phys. 458, Article ID 111121, 37 p. (2022; Zbl 07527733) Full Text: DOI arXiv
Yu, Jeremy; Lu, Lu; Meng, Xuhui; Karniadakis, George Em Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. (English) Zbl 1507.65217 Comput. Methods Appl. Mech. Eng. 393, Article ID 114823, 22 p. (2022). MSC: 65M99 65M32 68T07 PDFBibTeX XMLCite \textit{J. Yu} et al., Comput. Methods Appl. Mech. Eng. 393, Article ID 114823, 22 p. (2022; Zbl 1507.65217) Full Text: DOI arXiv
Lu, Lu; Meng, Xuhui; Cai, Shengze; Mao, Zhiping; Goswami, Somdatta; Zhang, Zhongqiang; Karniadakis, George Em A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. (English) Zbl 1507.65050 Comput. Methods Appl. Mech. Eng. 393, Article ID 114778, 35 p. (2022). MSC: 65D15 68T07 PDFBibTeX XMLCite \textit{L. Lu} et al., Comput. Methods Appl. Mech. Eng. 393, Article ID 114778, 35 p. (2022; Zbl 1507.65050) Full Text: DOI arXiv
Zhao, Yue; Mao, Zhiping; Guo, Ling; Tang, Yifa; Karniadakis, George Em A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition. (English) Zbl 07525181 J. Comput. Phys. 461, Article ID 111213, 17 p. (2022). MSC: 65M70 65C05 35R11 60H35 PDFBibTeX XMLCite \textit{Y. Zhao} et al., J. Comput. Phys. 461, Article ID 111213, 17 p. (2022; Zbl 07525181) Full Text: DOI
Lanthaler, Samuel; Mishra, Siddhartha; Karniadakis, George E. Error estimates for deeponets: a deep learning framework in infinite dimensions. (English) Zbl 07525076 Trans. Math. Appl. 6, No. 1, Article ID tnac001, 141 p. (2022). MSC: 68-XX 41-XX PDFBibTeX XMLCite \textit{S. Lanthaler} et al., Trans. Math. Appl. 6, No. 1, Article ID tnac001, 141 p. (2022; Zbl 07525076) Full Text: DOI arXiv
Meng, Xuhui; Yang, Liu; Mao, Zhiping; del Águila Ferrandis, José; Karniadakis, George Em Learning functional priors and posteriors from data and physics. (English) Zbl 1515.62046 J. Comput. Phys. 457, Article ID 111073, 22 p. (2022). MSC: 62F15 68T07 PDFBibTeX XMLCite \textit{X. Meng} et al., J. Comput. Phys. 457, Article ID 111073, 22 p. (2022; Zbl 1515.62046) Full Text: DOI arXiv
Cai, Min; Kharazmi, Ehsan; Li, Changpin; Karniadakis, George Em Fractional buffer layers: absorbing boundary conditions for wave propagation. (English) Zbl 1485.35376 Commun. Comput. Phys. 31, No. 2, 331-369 (2022). MSC: 35R11 35L05 26A33 65M70 76M22 PDFBibTeX XMLCite \textit{M. Cai} et al., Commun. Comput. Phys. 31, No. 2, 331--369 (2022; Zbl 1485.35376) Full Text: DOI arXiv
Goswami, Somdatta; Yin, Minglang; Yu, Yue; Karniadakis, George Em A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials. (English) Zbl 1507.74383 Comput. Methods Appl. Mech. Eng. 391, Article ID 114587, 29 p. (2022). MSC: 74R10 65N99 PDFBibTeX XMLCite \textit{S. Goswami} et al., Comput. Methods Appl. Mech. Eng. 391, Article ID 114587, 29 p. (2022; Zbl 1507.74383) Full Text: DOI arXiv
Yang, Liu; Daskalakis, Constantinos; Karniadakis, George E. Generative ensemble regression: Learning particle dynamics from observations of ensembles with physics-informed deep generative models. (English) Zbl 1491.60117 SIAM J. Sci. Comput. 44, No. 1, B80-B99 (2022). MSC: 60H35 60J22 62M05 PDFBibTeX XMLCite \textit{L. Yang} et al., SIAM J. Sci. Comput. 44, No. 1, B80--B99 (2022; Zbl 1491.60117) Full Text: DOI arXiv
Zhang, Qian; Kahana, Adar; Karniadakis, George Em; Stinis, Panos SMS: Spiking Marching Scheme for Efficient Long Time Integration of Differential Equations. arXiv:2211.09928 Preprint, arXiv:2211.09928 [math.NA] (2022). MSC: 65M99 BibTeX Cite \textit{Q. Zhang} et al., ``SMS: Spiking Marching Scheme for Efficient Long Time Integration of Differential Equations'', Preprint, arXiv:2211.09928 [math.NA] (2022) Full Text: arXiv OA License
Hu, Zheyuan; Jagtap, Ameya D.; Karniadakis, George Em; Kawaguchi, Kenji Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology. arXiv:2211.08939 Preprint, arXiv:2211.08939 [cs.LG] (2022). BibTeX Cite \textit{Z. Hu} et al., ``Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology'', Preprint, arXiv:2211.08939 [cs.LG] (2022) Full Text: DOI arXiv OA License
Zhang, Enrui; Kahana, Adar; Turkel, Eli; Ranade, Rishikesh; Pathak, Jay; Karniadakis, George Em A Hybrid Iterative Numerical Transferable Solver (HINTS) for PDEs Based on Deep Operator Network and Relaxation Methods. arXiv:2208.13273 Preprint, arXiv:2208.13273 [math.NA] (2022). BibTeX Cite \textit{E. Zhang} et al., ``A Hybrid Iterative Numerical Transferable Solver (HINTS) for PDEs Based on Deep Operator Network and Relaxation Methods'', Preprint, arXiv:2208.13273 [math.NA] (2022) Full Text: arXiv OA License
Goswami, Somdatta; Bora, Aniruddha; Yu, Yue; Karniadakis, George Em Physics-Informed Deep Neural Operator Networks. arXiv:2207.05748 Preprint, arXiv:2207.05748 [cs.LG] (2022). BibTeX Cite \textit{S. Goswami} et al., ``Physics-Informed Deep Neural Operator Networks'', Preprint, arXiv:2207.05748 [cs.LG] (2022) Full Text: arXiv OA License
Cai, Min; Karniadakis, George Em; Li, Changpin Fractional SEIR Model and Data-Driven Predictions of COVID-19 Dynamics of Omicron Variant. arXiv:2205.11379 Preprint, arXiv:2205.11379 [math.NA] (2022). BibTeX Cite \textit{M. Cai} et al., ``Fractional SEIR Model and Data-Driven Predictions of COVID-19 Dynamics of Omicron Variant'', Preprint, arXiv:2205.11379 [math.NA] (2022) Full Text: DOI arXiv OA License
Shukla, Khemraj; Xu, Mengjia; Trask, Nathaniel; Karniadakis, George Em Scalable algorithms for physics-informed neural and graph networks. arXiv:2205.08332 Preprint, arXiv:2205.08332 [cs.LG] (2022). BibTeX Cite \textit{K. Shukla} et al., ``Scalable algorithms for physics-informed neural and graph networks'', Preprint, arXiv:2205.08332 [cs.LG] (2022) Full Text: arXiv OA License
Howard, Amanda A.; Perego, Mauro; Karniadakis, George E.; Stinis, Panos Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems. arXiv:2204.09157 Preprint, arXiv:2204.09157 [math.NA] (2022). BibTeX Cite \textit{A. A. Howard} et al., ``Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems'', Preprint, arXiv:2204.09157 [math.NA] (2022) Full Text: DOI arXiv OA License
Jagtap, Ameya D.; Mitsotakis, Dimitrios; Karniadakis, George Em Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations. arXiv:2202.02899 Preprint, arXiv:2202.02899 [physics.flu-dyn] (2022). BibTeX Cite \textit{A. D. Jagtap} et al., ``Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations'', Preprint, arXiv:2202.02899 [physics.flu-dyn] (2022) Full Text: arXiv OA License
Carlson, Max; Zheng, Xiaoning; Sundar, Hari; Karniadakis, George Em; Kirby, Robert M. An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains. (English) Zbl 07690829 Comput. Phys. Commun. 261, Article ID 107695, 11 p. (2021). MSC: 65-XX 74-XX PDFBibTeX XMLCite \textit{M. Carlson} et al., Comput. Phys. Commun. 261, Article ID 107695, 11 p. (2021; Zbl 07690829) Full Text: DOI
Mao, Zhiping; Lu, Lu; Marxen, Olaf; Zaki, Tamer A.; Karniadakis, George Em DeepM&Mnet for hypersonics: predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators. (English) Zbl 07516442 J. Comput. Phys. 447, Article ID 110698, 24 p. (2021). MSC: 76Mxx 76Fxx 65Mxx PDFBibTeX XMLCite \textit{Z. Mao} et al., J. Comput. Phys. 447, Article ID 110698, 24 p. (2021; Zbl 07516442) Full Text: DOI arXiv
Shukla, Khemraj; Jagtap, Ameya D.; Karniadakis, George Em Parallel physics-informed neural networks via domain decomposition. (English) Zbl 07516435 J. Comput. Phys. 447, Article ID 110683, 19 p. (2021). MSC: 68Txx 65Yxx 65Mxx PDFBibTeX XMLCite \textit{K. Shukla} et al., J. Comput. Phys. 447, Article ID 110683, 19 p. (2021; Zbl 07516435) Full Text: DOI arXiv
Lou, Qin; Meng, Xuhui; Karniadakis, George Em Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation. (English) Zbl 07516434 J. Comput. Phys. 447, Article ID 110676, 20 p. (2021). MSC: 76Mxx 76Pxx 68Txx PDFBibTeX XMLCite \textit{Q. Lou} et al., J. Comput. Phys. 447, Article ID 110676, 20 p. (2021; Zbl 07516434) Full Text: DOI arXiv
Cai, Shengze; Wang, Zhicheng; Lu, Lu; Zaki, Tamer A.; Karniadakis, George Em DeepM&Mnet: inferring the electroconvection multiphysics fields based on operator approximation by neural networks. (English) Zbl 07513856 J. Comput. Phys. 436, Article ID 110296, 17 p. (2021). MSC: 76Mxx 68Txx 76Fxx PDFBibTeX XMLCite \textit{S. Cai} et al., J. Comput. Phys. 436, Article ID 110296, 17 p. (2021; Zbl 07513856) Full Text: DOI arXiv
Zhao, Lifei; Li, Zhen; Wang, Zhicheng; Caswell, Bruce; Ouyang, Jie; Karniadakis, George Em Active- and transfer-learning applied to microscale-macroscale coupling to simulate viscoelastic flows. (English) Zbl 07510254 J. Comput. Phys. 427, Article ID 110069, 18 p. (2021). MSC: 76-XX 74-XX PDFBibTeX XMLCite \textit{L. Zhao} et al., J. Comput. Phys. 427, Article ID 110069, 18 p. (2021; Zbl 07510254) Full Text: DOI arXiv
Jin, Xiaowei; Cai, Shengze; Li, Hui; Karniadakis, George Em NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations. (English) Zbl 07510065 J. Comput. Phys. 426, Article ID 109951, 26 p. (2021). MSC: 76-XX 65-XX PDFBibTeX XMLCite \textit{X. Jin} et al., J. Comput. Phys. 426, Article ID 109951, 26 p. (2021; Zbl 07510065) Full Text: DOI arXiv
Yang, Liu; Meng, Xuhui; Karniadakis, George Em B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. (English) Zbl 07508507 J. Comput. Phys. 425, Article ID 109913, 23 p. (2021). MSC: 35-XX 62-XX PDFBibTeX XMLCite \textit{L. Yang} et al., J. Comput. Phys. 425, Article ID 109913, 23 p. (2021; Zbl 07508507) Full Text: DOI arXiv
Meng, Xuhui; Babaee, Hessam; Karniadakis, George Em Multi-fidelity Bayesian neural networks: algorithms and applications. (English) Zbl 07505956 J. Comput. Phys. 438, Article ID 110361, 16 p. (2021). MSC: 68Txx 65Cxx 76-XX PDFBibTeX XMLCite \textit{X. Meng} et al., J. Comput. Phys. 438, Article ID 110361, 16 p. (2021; Zbl 07505956) Full Text: DOI arXiv
Wang, Zhicheng; Zheng, Xiaoning; Chryssostomidis, Chryssostomos; Karniadakis, George Em A phase-field method for boiling heat transfer. (English) Zbl 07503725 J. Comput. Phys. 435, Article ID 110239, 19 p. (2021). MSC: 76Mxx 76Txx 76Dxx PDFBibTeX XMLCite \textit{Z. Wang} et al., J. Comput. Phys. 435, Article ID 110239, 19 p. (2021; Zbl 07503725) Full Text: DOI
D’Elia, Marta; Gulian, Mamikon; Olson, Hayley; Karniadakis, George Em Towards a unified theory of fractional and nonlocal vector calculus. (English) Zbl 1498.26008 Fract. Calc. Appl. Anal. 24, No. 5, 1301-1355 (2021). MSC: 26A33 35R11 34A08 PDFBibTeX XMLCite \textit{M. D'Elia} et al., Fract. Calc. Appl. Anal. 24, No. 5, 1301--1355 (2021; Zbl 1498.26008) Full Text: DOI arXiv
Chen, Xiaoli; Duan, Jinqiao; Karniadakis, George Em Learning and meta-learning of stochastic advection-diffusion-reaction systems from sparse measurements. (English) Zbl 07441294 Eur. J. Appl. Math. 32, No. 3, 397-420 (2021). MSC: 68T05 35R30 PDFBibTeX XMLCite \textit{X. Chen} et al., Eur. J. Appl. Math. 32, No. 3, 397--420 (2021; Zbl 07441294) Full Text: DOI arXiv
Blumers, Ansel L.; Yin, Minglang; Nakajima, Hiroyuki; Hasegawa, Yosuke; Li, Zhen; Karniadakis, George Em Multiscale parareal algorithm for long-time mesoscopic simulations of microvascular blood flow in zebrafish. (English) Zbl 1479.76123 Comput. Mech. 68, No. 5, 1131-1152 (2021). MSC: 76Z05 76D05 76A05 76M28 92C35 PDFBibTeX XMLCite \textit{A. L. Blumers} et al., Comput. Mech. 68, No. 5, 1131--1152 (2021; Zbl 1479.76123) Full Text: DOI arXiv
Meng, Xuhui; Wang, Zhicheng; Fan, Dixia; Triantafyllou, Michael S.; Karniadakis, George Em A fast multi-fidelity method with uncertainty quantification for complex data correlations: application to vortex-induced vibrations of marine risers. (English) Zbl 1507.62298 Comput. Methods Appl. Mech. Eng. 386, Article ID 114212, 19 p. (2021). MSC: 62M45 65C99 70J10 76B47 PDFBibTeX XMLCite \textit{X. Meng} et al., Comput. Methods Appl. Mech. Eng. 386, Article ID 114212, 19 p. (2021; Zbl 1507.62298) Full Text: DOI arXiv
Lin, Chensen; Maxey, Martin; Li, Zhen; Karniadakis, George Em A seamless multiscale operator neural network for inferring bubble dynamics. (English) Zbl 1495.76119 J. Fluid Mech. 929, Paper No. A18, 14 p. (2021). MSC: 76T10 76M99 76M28 68T05 PDFBibTeX XMLCite \textit{C. Lin} et al., J. Fluid Mech. 929, Paper No. A18, 14 p. (2021; Zbl 1495.76119) Full Text: DOI
Chen, Xiaoli; Yang, Liu; Duan, Jinqiao; Karniadakis, George Em Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks. (English) Zbl 1480.35377 SIAM J. Sci. Comput. 43, No. 3, B811-B830 (2021). Reviewer: Wasiur Rahman Khuda Bukhsh (Nottingham) MSC: 35Q84 62M45 60H35 60J65 35B30 92B20 PDFBibTeX XMLCite \textit{X. Chen} et al., SIAM J. Sci. Comput. 43, No. 3, B811--B830 (2021; Zbl 1480.35377) Full Text: DOI arXiv
Cai, Shengze; Wang, Zhicheng; Fuest, Frederik; Jeon, Young Jin; Gray, Callum; Karniadakis, George Em Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. (English) Zbl 1461.76417 J. Fluid Mech. 915, Paper No. A102, 17 p. (2021). MSC: 76R05 PDFBibTeX XMLCite \textit{S. Cai} et al., J. Fluid Mech. 915, Paper No. A102, 17 p. (2021; Zbl 1461.76417) Full Text: DOI arXiv
Yin, Minglang; Zheng, Xiaoning; Humphrey, Jay D.; Karniadakis, George Em Non-invasive inference of thrombus material properties with physics-informed neural networks. (English) Zbl 1506.74215 Comput. Methods Appl. Mech. Eng. 375, Article ID 113603, 23 p. (2021). MSC: 74L15 PDFBibTeX XMLCite \textit{M. Yin} et al., Comput. Methods Appl. Mech. Eng. 375, Article ID 113603, 23 p. (2021; Zbl 1506.74215) Full Text: DOI arXiv
Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M. hp-VPINNs: variational physics-informed neural networks with domain decomposition. (English) Zbl 1506.68105 Comput. Methods Appl. Mech. Eng. 374, Article ID 113547, 25 p. (2021). MSC: 68T07 92B20 65M55 PDFBibTeX XMLCite \textit{E. Kharazmi} et al., Comput. Methods Appl. Mech. Eng. 374, Article ID 113547, 25 p. (2021; Zbl 1506.68105) Full Text: DOI arXiv
Clark Di Leoni, Patricio; Zaki, Tamer A.; Karniadakis, George; Meneveau, Charles Two-point stress-strain-rate correlation structure and non-local eddy viscosity in turbulent flows. (English) Zbl 1461.76181 J. Fluid Mech. 914, Paper No. A6, 20 p. (2021). MSC: 76F02 76F55 PDFBibTeX XMLCite \textit{P. Clark Di Leoni} et al., J. Fluid Mech. 914, Paper No. A6, 20 p. (2021; Zbl 1461.76181) Full Text: DOI arXiv
Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em DeepXDE: a deep learning library for solving differential equations. (English) Zbl 1459.65002 SIAM Rev. 63, No. 1, 208-228 (2021). MSC: 65-01 65-04 68T07 65L99 65M99 65N99 PDFBibTeX XMLCite \textit{L. Lu} et al., SIAM Rev. 63, No. 1, 208--228 (2021; Zbl 1459.65002) Full Text: DOI arXiv
Zhang, Zhen; Shin, Yeonjong; Karniadakis, George Em GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems. arXiv:2109.00092 Preprint, arXiv:2109.00092 [math.DS] (2021). BibTeX Cite \textit{Z. Zhang} et al., ``GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems'', Preprint, arXiv:2109.00092 [math.DS] (2021) Full Text: DOI arXiv OA License
Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em A Caputo fractional derivative-based algorithm for optimization. arXiv:2104.02259 Preprint, arXiv:2104.02259 [math.OC] (2021). MSC: 65K05 65B99 26A33 BibTeX Cite \textit{Y. Shin} et al., ``A Caputo fractional derivative-based algorithm for optimization'', Preprint, arXiv:2104.02259 [math.OC] (2021) Full Text: arXiv OA License
Deng, Beichuan; Shin, Yeonjong; Lu, Lu; Zhang, Zhongqiang; Karniadakis, George Em Convergence rate of DeepONets for learning operators arising from advection-diffusion equations. arXiv:2102.10621 Preprint, arXiv:2102.10621 [math.AP] (2021). BibTeX Cite \textit{B. Deng} et al., ``Convergence rate of DeepONets for learning operators arising from advection-diffusion equations'', Preprint, arXiv:2102.10621 [math.AP] (2021) Full Text: arXiv OA License
Pang, G.; D’Elia, M.; Parks, M.; Karniadakis, G. E. nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications. (English) Zbl 07508384 J. Comput. Phys. 422, Article ID 109760, 26 p. (2020). MSC: 35-XX 86-XX PDFBibTeX XMLCite \textit{G. Pang} et al., J. Comput. Phys. 422, Article ID 109760, 26 p. (2020; Zbl 07508384) Full Text: DOI arXiv
Zheng, Qiang; Zeng, Lingzao; Karniadakis, George Em Physics-informed semantic inpainting: application to geostatistical modeling. (English) Zbl 07507237 J. Comput. Phys. 419, Article ID 109676, 12 p. (2020). MSC: 86-XX 68-XX PDFBibTeX XMLCite \textit{Q. Zheng} et al., J. Comput. Phys. 419, Article ID 109676, 12 p. (2020; Zbl 07507237) Full Text: DOI arXiv
Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. (English) Zbl 1478.76057 Science 367, No. 6481, 1026-1030 (2020). MSC: 76M27 68T07 92C35 76Z05 76D05 PDFBibTeX XMLCite \textit{M. Raissi} et al., Science 367, No. 6481, 1026--1030 (2020; Zbl 1478.76057) Full Text: DOI Link
Jin, Pengzhan; Zhang, Zhen; Zhu, Aiqing; Tang, Yifa; Karniadakis, George Em Sympnets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. (English) Zbl 1475.68316 Neural Netw. 132, 166-179 (2020). MSC: 68T07 70H99 PDFBibTeX XMLCite \textit{P. Jin} et al., Neural Netw. 132, 166--179 (2020; Zbl 1475.68316) Full Text: DOI arXiv
Jin, Pengzhan; Lu, Lu; Tang, Yifa; Karniadakis, George Em Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness. (English) Zbl 1475.68315 Neural Netw. 130, 85-99 (2020). MSC: 68T07 PDFBibTeX XMLCite \textit{P. Jin} et al., Neural Netw. 130, 85--99 (2020; Zbl 1475.68315) Full Text: DOI arXiv
Shin, Yeonjong; Darbon, Jérôme; Karniadakis, George Em On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. (English) Zbl 1473.65349 Commun. Comput. Phys. 28, No. 5, 2042-2074 (2020). MSC: 65N99 65M99 65M12 65N12 35J25 35K20 68T07 PDFBibTeX XMLCite \textit{Y. Shin} et al., Commun. Comput. Phys. 28, No. 5, 2042--2074 (2020; Zbl 1473.65349) Full Text: DOI arXiv
Jagtap, Ameya D.; Karniadakis, George Em Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. (English) Zbl 07419158 Commun. Comput. Phys. 28, No. 5, 2002-2041 (2020). MSC: 65N55 65M55 65N21 35E05 35E15 76L05 74B05 68T07 68Q32 35J05 35Q35 PDFBibTeX XMLCite \textit{A. D. Jagtap} and \textit{G. E. Karniadakis}, Commun. Comput. Phys. 28, No. 5, 2002--2041 (2020; Zbl 07419158) Full Text: DOI
Lu, Lu; Shin, Yeonjong; Su, Yanhui; Karniadakis, George Em Dying ReLU and initialization: theory and numerical examples. (English) Zbl 1507.68248 Commun. Comput. Phys. 28, No. 5, 1671-1706 (2020). MSC: 68T05 68T07 62M45 PDFBibTeX XMLCite \textit{L. Lu} et al., Commun. Comput. Phys. 28, No. 5, 1671--1706 (2020; Zbl 1507.68248) Full Text: DOI arXiv
Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. (English) Zbl 1472.68175 Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2239, Article ID 20200334, 20 p. (2020). MSC: 68T07 00A79 PDFBibTeX XMLCite \textit{A. D. Jagtap} et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2239, Article ID 20200334, 20 p. (2020; Zbl 1472.68175) Full Text: DOI arXiv Link
Meng, Xuhui; Li, Zhen; Zhang, Dongkun; Karniadakis, George Em PPINN: parareal physics-informed neural network for time-dependent PDEs. (English) Zbl 1506.65181 Comput. Methods Appl. Mech. Eng. 370, Article ID 113250, 16 p. (2020). MSC: 65M99 65Y05 68T07 PDFBibTeX XMLCite \textit{X. Meng} et al., Comput. Methods Appl. Mech. Eng. 370, Article ID 113250, 16 p. (2020; Zbl 1506.65181) Full Text: DOI arXiv
Zhang, Hui; Jiang, Xiaoyun; Zeng, Fanhai; Karniadakis, George Em A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations. (English) Zbl 1453.65370 J. Comput. Phys. 405, Article ID 109141, 17 p. (2020). MSC: 65M70 65M15 35R11 65M12 PDFBibTeX XMLCite \textit{H. Zhang} et al., J. Comput. Phys. 405, Article ID 109141, 17 p. (2020; Zbl 1453.65370) Full Text: DOI arXiv
Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. (English) Zbl 1453.68165 J. Comput. Phys. 404, Article ID 109136, 23 p. (2020). MSC: 68T07 65D15 65M32 PDFBibTeX XMLCite \textit{A. D. Jagtap} et al., J. Comput. Phys. 404, Article ID 109136, 23 p. (2020; Zbl 1453.68165) Full Text: DOI arXiv
Lischke, Anna; Pang, Guofei; Gulian, Mamikon; Song, Fangying; Glusa, Christian; Zheng, Xiaoning; Mao, Zhiping; Cai, Wei; Meerschaert, Mark M.; Ainsworth, Mark; Karniadakis, George Em What is the fractional Laplacian? A comparative review with new results. (English) Zbl 1453.35179 J. Comput. Phys. 404, Article ID 109009, 62 p. (2020). MSC: 35R11 60G51 35A01 35A02 65N30 65C05 35-02 65-02 PDFBibTeX XMLCite \textit{A. Lischke} et al., J. Comput. Phys. 404, Article ID 109009, 62 p. (2020; Zbl 1453.35179) Full Text: DOI Link
Meng, Xuhui; Karniadakis, George Em A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. (English) Zbl 1454.76006 J. Comput. Phys. 401, Article ID 109020, 15 p. (2020). MSC: 76-10 68T07 76S05 76V05 PDFBibTeX XMLCite \textit{X. Meng} and \textit{G. E. Karniadakis}, J. Comput. Phys. 401, Article ID 109020, 15 p. (2020; Zbl 1454.76006) Full Text: DOI arXiv
Kharazmi, Ehsan; Mao, Zhiping; Pang, Guofei; Zayernouri, Mohsen; Karniadakis, George Em Fractional calculus and numerical methods for fractional PDEs. (English) Zbl 1442.35514 Emmanouil, Ioannis (ed.) et al., First congress of Greek mathematicians. Proceedings of the congress held in Athens, Greece, June 25–30, 2018. Berlin: De Gruyter. De Gruyter Proc. Math., 91-125 (2020). MSC: 35R11 65M70 PDFBibTeX XMLCite \textit{E. Kharazmi} et al., in: First congress of Greek mathematicians. Proceedings of the congress held in Athens, Greece, June 25--30, 2018. Berlin: De Gruyter. 91--125 (2020; Zbl 1442.35514) Full Text: DOI
Li, Xianjuan; Mao, Zhiping; Wang, Nan; Song, Fangying; Wang, Hong; Karniadakis, George Em A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation. (English) Zbl 1442.65144 Comput. Methods Appl. Mech. Eng. 366, Article ID 113053, 24 p. (2020). MSC: 65L60 34A08 PDFBibTeX XMLCite \textit{X. Li} et al., Comput. Methods Appl. Mech. Eng. 366, Article ID 113053, 24 p. (2020; Zbl 1442.65144) Full Text: DOI arXiv