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Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects.
Universal approximation bounds for superpositions of a sigmoidal function
Andrew R Barron · 1993
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Tempered stable lévy motion and transient super-diffusion
Boris Baeumer and Mark M. Meerschaert · 2010
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Fractional calculus for power functions and eigenvalues of the fractional Laplacian
Bartłlomiej Dyda · 2012
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A Chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation
Emmanuel Hanert and Cécile Piret · 2014
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Adam: A method for stochastic optimization
Diederik P Kingma and Jimmy Ba · 2014
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Tempered fractional calculus
Farzad Sabzikar, Mark M. Meerschaert, and Jinghua Chen · 2015
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Fokker–Planck equations for stochastic dynamical systems with symmetric lévy motions
Ting Gao, Jinqiao Duan, and Xiaofan Li · 2016
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High order schemes for the tempered fractional diffusion equations
Can Li and Weihua Deng · 2016
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Spectral methods for tempered fractional differential equations
Lijing Zhao, Weihua Deng, and Jan S Hesthaven · 2016
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Generalization in deep learning
Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio · 2017
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Boundary problems for the fractional and tempered fractional operators
Weihua Deng, Buyang Li, Wenyi Tian, and Pingwen Zhang · 2018
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Numerical approximations for the tempered fractional Laplacian: Error analysis and applications
Siwei Duo and Yanzhi Zhang · 2019
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Handbook of fractional calculus with applications
George Em Karniadakis, editor · 2019
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fPINNs: Fractional physics-informed neural networks
Guofei Pang, Lu Lu, and George Em Karniadakis · 2019
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Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
Maziar Raissi, Paris Perdikaris, and George E Karniadakis · 2019
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Numerical methods for nonlocal and fractional models
Marta D’Elia, Qiang Du, Christian Glusa, Max Gunzburger, Xiaochuan Tian, and Zhi Zhou · 2020
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What is the fractional Laplacian? a comparative review with new results
Anna Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth, and George Em Karniadakis · 2020
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Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations
Ling Guo, Hao Wu, Xiaochen Yu, and Tao Zhou · 2022
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When do extended physics-informed neural networks (XPINNs) improve generalization?
Zheyuan Hu, Ameya D Jagtap, George Em Karniadakis, and Kenji Kawaguchi · 2022
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Monte Carlo method for fractional-order differentiation extended to higher orders
Nikolai Leonenko and Igor Podlubny · 2022
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deepfdenet: A novel neural network architecture for solving fractional differential equations
Ali Nosrati Firoozsalari, Hassan Dana Mazraeh, Alireza Afzal Aghaei, and Kourosh Parand · 2023
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Tao Luo and H. Yang · 2020
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Siddhartha Mishra and Roberto Molinaro · 2020
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Solving inverse stochastic problems from discrete particle observations using the Fokker–Planck equation and physics-informed neural networks
Xiaoli Chen, Liu Yang, Jinqiao Duan, and George Em Karniadakis · 2021
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Tempered and hadamard-type fractional calculus with respect to functions
Hafiz Muhammad Fahad, Arran Fernandez, Mujeeb Ur Rehman, and Maham Siddiqi · 2021
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Physics-informed neural networks with hard constraints for inverse design
Lu Lu, Raphael Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G Johnson · 2021
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Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian
Jing Sun, Daxin Nie, and Weihua Deng · 2021
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Error estimates for physics informed neural networks approximating the navier-stokes equations
Tim De Ryck, Ameya D Jagtap, and Siddhartha Mishra · 2022
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Zheyuan Hu, Zhouhao Yang, Yezhen Wang, George Em Karniadakis, and Kenji Kawaguchi · 2023
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Lei Ma, Fanhai Zeng, Ling Guo, George Em Karniadakis, et al · 2023
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Efficient Monte Carlo method for integral fractional Laplacian in multiple dimensions
Changtao Sheng, Bihao Su, and Chenglong Xu · 2023
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Hutchinson trace estimation for high-dimensional and high-order physics-informed neural networks
Zheyuan Hu, Zekun Shi, George Em Karniadakis, and Kenji Kawaguchi · 2024
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Tackling the curse of dimensionality with physics-informed neural networks
Zheyuan Hu, Khemraj Shukla, George Em Karniadakis, and Kenji Kawaguchi · 2024
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Score-based physics-informed neural networks for high-dimensional Fokker-Planck equations
Zheyuan Hu, Zhongqiang Zhang, George Em Karniadakis, and Kenji Kawaguchi · 2024
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Shupeng Wang and George Em Karniadakis · 2024
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