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Fourier Neural Operators (FNO) offer a principled approach to solving challenging partial differential equations (PDE) such as turbulent flows.
The convergence rate of neural networks for learned functions of different frequencies
Basri, R., Jacobs, D., Kasten, Y., and Kritchman, S · 1906
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Splitting steepest descent for growing neural architectures
Liu, Q., Wu, L., and Wang, D · 1910
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Towards understanding the spectral bias of deep learning
Cao, Y., Fang, Z., Wu, Y., Zhou, D.-X., and Gu, Q · 1912
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Neural-network-based approximations for solving partial differential equations
Dissanayake, M. and Phan-Thien, N · 1994
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Artificial neural networks for solving ordinary and partial differential equations
Lagaris, I. E., Likas, A., and Fotiadis, D. I · 1998
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Neural operator: Graph kernel network for partial differential equations, b
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A · 2003
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Computing nearly singular solutions using pseudo-spectral methods
Hou, T. Y. and Li, R · 2007
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Fourier neural operator for parametric partial differential equations, a
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A · 2010
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Two-dimensional turbulence
Boffetta, G., Ecke, R. E., et al · 2012
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Convolutional neural networks for steady flow approximation
Guo, X., Li, W., and Iorio, F · 2016
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Cyclical learning rates for training neural networks
Smith, L. N · 2017
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Dgm: A deep learning algorithm for solving partial differential equations
Sirignano, J. and Spiliopoulos, K · 2018
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The deep ritz method: a deep learning-based numerical algorithm for solving variational problems
Weinan, E. and Yu, B · 2018
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Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification
Zhu, Y. and Zabaras, N · 2018
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Prediction of aerodynamic flow fields using convolutional neural networks
Bhatnagar, S., Afshar, Y., Pan, S., Duraisamy, K., and Kaushik, S · 2019
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Learning to optimize multigrid pde solvers
Greenfeld, D., Galun, M., Basri, R., Yavneh, I., and Kimmel, R · 2019
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Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
Learning nonlinear operators via deeponet based on the universal approximation theorem of operators
Lu, L., Jin, P., Pang, G., Zhang, Z., and Karniadakis, G. E · 2021
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The random feature model for input-output maps between banach spaces
Nelsen, N. H. and Stuart, A. M · 2021
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Ml-pde: A framework for a machine learning enhanced pde solver
Pathak, J., Mustafa, M., Kashinath, K., Motheau, E., Kurth, T., and Day, M · 2021
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Towards large-scale learned solvers for parametric pdes with model-parallel fourier neural operators
Grady II, T. J., Khan, R., Louboutin, M., Yin, Z., Witte, P. A., Chandra, R., Hewett, R. J., and Herrmann, F. J · 2022
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Learning operators with coupled attention
Kissas, G., Seidman, J. H., Guilhoto, L. F., Preciado, V. M., Pappas, G. J., and Perdikaris, P · 2022
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Raissi, M., Perdikaris, P., and Karniadakis, G. E · 2019
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Frequency principle: Fourier analysis sheds light on deep neural networks
Xu, Z.-Q. J., Zhang, Y., Luo, T., Xiao, Y., and Ma, Z · 2019
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Machine learning accelerated computational fluid dynamics
Kochkov, D., Smith, J. A., Alieva, A., Wang, Q., Brenner, M. P., and Hoyer, S · 2021
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Neural operator: Learning maps between function spaces
Kovachki, N., Li, Z., Liu, B., Azizzadenesheli, K., Bhattacharya, K., Stuart, A., and Anandkumar, A · 2021
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Characterizing possible failure modes in physics-informed neural networks
Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and Mahoney, M. W · 2021
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Markov neural operators for learning chaotic systems
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A · 2021
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Spectral bias in practice: The role of function frequency in generalization
Fridovich-Keil, S., Gontijo-Lopes, R., and Roelofs, R
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A learning-based multiscale method and its application to inelastic impact problems
Liu, B., Kovachki, N., Li, Z., Azizzadenesheli, K., Anandkumar, A., Stuart, A. M., and Bhattacharya, K · 2022
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Pathak, J., Subramanian, S., Harrington, P., Raja, S., Chattopadhyay, A., Mardani, M., Kurth, T., Hall, D., Li, Z., Azizzadenesheli, K., et al · 2022
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Curriculum learning: A survey
Soviany, P., Ionescu, R. T., Rota, P., and Sebe, N · 2022
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Pdebench: An extensive benchmark for scientific machine learning
Takamoto, M., Praditia, T., Leiteritz, R., MacKinlay, D., Alesiani, F., Pflüger, D., and Niepert, M · 2022
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U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow
Wen, G., Li, Z., Azizzadenesheli, K., Anandkumar, A., and Benson, S. M · 2022
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Multi-grid tensorized fourier neural operator for high-resolution pdes
Kossaifi, J., Kovachki, N., Azizzadenesheli, K., and Anandkumar, A · 2023
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