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We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE).
Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems
T. Chen and H. Chen · 1995
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Gaussian processes for machine learning
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U-net: Convolutional networks for biomedical image segmentation
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Deep unsupervised learning using nonequilibrium thermodynamics
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Y. Zhu, N. Zabaras, P.-S. Koutsourelakis, and P. Perdikaris · 2019
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Denoising diffusion probabilistic models
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Fourier neural operator for parametric partial differential equations
Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar · 2020
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Neural operator: Graph kernel network for partial differential equations
Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar · 2020
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Denoising diffusion implicit models
J. Song, C. Meng, and S. Ermon · 2020
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Score-based generative modeling through stochastic differential equations
Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole · 2020
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Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data
L. Sun, H. Gao, S. Pan, and J.-X. Wang · 2020
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Towards physics-informed deep learning for turbulent flow prediction
R. Wang, K. Kashinath, M. Mustafa, A. Albert, and R. Yu · 2020
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Model reduction and neural networks for parametric pdes
K. Bhattacharya, B. Hosseini, N. B. Kovachki, and A. M. Stuart · 2021
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Understanding and mitigating gradient flow pathologies in physics-informed neural networks
S. Wang, Y. Teng, and P. Perdikaris · 2021
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Learning the solution operator of parametric partial differential equations with physics-informed deeponets
S. Wang, H. Wang, and P. Perdikaris · 2021
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The cost-accuracy trade-off in operator learning with neural networks
M. De Hoop, D. Z. Huang, E. Qian, and A. M. Stuart · 2022
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Cascaded diffusion models for high fidelity image generation
J. Ho, C. Saharia, W. Chan, D. J. Fleet, M. Norouzi, and T. Salimans · 2022
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Understanding ddpm latent codes through optimal transport
V. Khrulkov, G. Ryzhakov, A. Chertkov, and I. Oseledets · 2022
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Diffusion models beat gans on image synthesis
P. Dhariwal and A. Nichol · 2021
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Multiwavelet-based operator learning for differential equations
G. Gupta, X. Xiao, and P. Bogdan · 2021
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On universal approximation and error bounds for Fourier neural operators
N. Kovachki, S. Lanthaler, and S. Mishra · 2021
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Neural operator: Learning maps between function spaces
N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandkumar · 2021
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Physics-informed neural operator for learning partial differential equations
Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, and A. Anandkumar · 2021
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A physics-informed operator regression framework for extracting data-driven continuum models
R. G. Patel, N. A. Trask, M. A. Wood, and E. C. Cyr · 2021
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A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data
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A. Phillips, T. Seror, M. Hutchinson, V. De Bortoli, A. Doucet, and E. Mathieu · 2022
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High-resolution image synthesis with latent diffusion models
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When and why pinns fail to train: A neural tangent kernel perspective
S. Wang, X. Yu, and P. Perdikaris · 2022
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Stochastic deep-ritz for parametric uncertainty quantification
T. Wang and J. Knap · 2022
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Kernel methods are competitive for operator learning
P. Batlle, M. Darcy, B. Hosseini, and H. Owhadi · 2023
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