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We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning.
The wave of advance of advantageous genes
R. A. Fisher · 1937
Earlier work this paper cites.
Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique
A. N. Kolmogorov · 1937
Earlier work this paper cites.
Application of brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov
H. P. McKean · 1975
Earlier work this paper cites.
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening
S. M. Allen and J. W. Cahn · 1979
Earlier work this paper cites.
Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
Earlier work this paper cites.
Neural-network-based approximations for solving partial differential equations
M. Dissanayake and N. Phan-Thien · 1994
Earlier work this paper cites.
Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems
T. Chen and H. Chen · 1995
Earlier work this paper cites.
A multivariate Faa di Bruno formula with applications
G. Constantine and T. Savits · 1996
Earlier work this paper cites.
Neural-network methods for boundary value problems with irregular boundaries
I. E. Lagaris, A. Likas, and P. G. D · 2000
Earlier work this paper cites.
Artificial neural networks for solving ordinary and partial differential equations
I. E. Lagaris, A. Likas, and D. I. Fotiadis · 2000
Earlier work this paper cites.
Stochastic differential equations
B. Øksendal · 2003
Earlier work this paper cites.
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs
A. Cohen, R. Devore, and C. Schwab · 2011
Earlier work this paper cites.
Counterparty risk valuation: A marked branching diffusion approach
P. Henry-Labordere · 2012
Earlier work this paper cites.
A numerical algorithm for a class of BSDEs via the branching process
P. Henry-Labordere, X. Tan, and N. Touzi · 2014
Earlier work this paper cites.
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
W. E, J. Han, and A. Jentzen · 2017
Earlier work this paper cites.
Error bounds for approximations with deep ReLU networks
D. Yarotsky · 2017
Earlier work this paper cites.
Numerical Analysis of Stochastic Ordinary Differential Equations
A. Barth, A. Jentzen, A. Lang, and C. Schwab · 2018
Earlier work this paper cites.
The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems
W. E and B. Yu · 2018
Earlier work this paper cites.
P. Grohs, F. Hornung, A. Jentzen, and P. Von Wurstemberger · 2018
Earlier work this paper cites.
A. Jentzen, D. Salimova, and T. Welti · 2018
Earlier work this paper cites.
Hidden physics models: Machine learning of nonlinear partial differential equations
M. Raissi and G. E. Karniadakis · 2018
Earlier work this paper cites.
M. Raissi, A. Yazdani, and G. E. Karniadakis · 2018
Earlier work this paper cites.
Space-time error estimates for deep neural network approximations for differential equations
P. Grohs, F. Hornung, A. Jentzen, and P. Zimmermann · 2019
Earlier work this paper cites.
W. H. Guss and R. Salakhutdinov · 2019
Earlier work this paper cites.
fPINNs: Fractional physics-informed neural networks
G. Pang, L. Lu, and G. E. Karniadakis · 2019
Earlier work this paper cites.
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
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Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in uq
C. Schwab and J. Zech · 2019
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C. Beck, L. Gonon, and A. Jentzen · 2020
Cited alongside, same era.
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
C. Beck, F. Hornung, M. Hutzenthaler, A. Jentzen, and T. Kruse · 2020
Cited alongside, same era.
Full error analysis for the training of deep neural networks, 2020
Approximation rates for neural networks with encodable weights in smoothness spaces
I. Gühring and M. Raslan · 2021
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On universal approximation and error bounds for Fourier Neural Operators
N. Kovachki, S. Lanthaler, and S. Mishra · 2021
Later among the works it cites.
Neural operator: Learning maps between function spaces
N. Kovachki, Z. Li, B. Liu, K. Azizzadensheli, K. Bhattacharya, A. Stuart, and A. Anandkumar · 2021
Later among the works it cites.
A theoretical analysis of deep neural networks and parametric PDEs
G. Kutyniok, P. Petersen, M. Raslan, and R. Schneider · 2021
Later among the works it cites.
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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C. Beck, A. Jentzen, and B. Kuckuck · 2020
Cited alongside, same era.
Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
J. Berner, P. Grohs, and A. Jentzen · 2020
Cited alongside, same era.
Error bounds for approximations with deep ReLU neural networks in
I. Gühring, G. Kutyniok, and P. Petersen · 2020
Cited alongside, same era.
Space-time deep neural network approximations for high-dimensional partial differential equations
F. Hornung, A. Jentzen, and D. Salimova · 2020
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A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
M. Hutzenthaler, A. Jentzen, T. Kruse, and T. A. Nguyen · 2020
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Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations
A. D. Jagtap and G. E. Karniadakis · 2020
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Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems
A. D. Jagtap, E. Kharazmi, and G. E. Karniadakis · 2020
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Fourier neural operator for parametric partial differential equations, 2020
Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar · 2020
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Operator learning for predicting multiscale bubble growth dynamics
C. Lin, Z. Li, L. Lu, S. Cai, M. Maxey, and G. E. Karniadakis · 2021
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Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis · 2021
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DeepXDE: A deep learning library for solving differential equations
L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis · 2021
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A priori generalization analysis of the deep Ritz method for solving high dimensional elliptic partial differential equations
Y. Lu, J. Lu, and M. Wang · 2021
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DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators
Z. Mao, L. Lu, O. Marxen, T. A. Zaki, and G. E. Karniadakis · 2021
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Parametric complexity bounds for approximating pdes with neural networks
T. Marwah, Z. Lipton, and A. Risteski · 2021
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Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs
S. Mishra and R. Molinaro · 2021
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Physics informed neural networks for simulating radiative transfer
S. Mishra and R. Molinaro · 2021
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Exponential ReLU DNN expression of holomorphic maps in high dimension
J. A. Opschoor, C. Schwab, and J. Zech · 2021
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Long-time integration of parametric evolution equations with physics-informed DeepONets
S. Wang and P. Perdikaris · 2021
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S. Wang, H. Wang, and P. Perdikaris · 2021
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B-PINNs: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data
L. Yang, X. Meng, and G. E. Karniadakis · 2021
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Error estimates for physics informed neural networks approximating the Navier-Stokes equations
T. De Ryck, A. D. Jagtap, and S. Mishra · 2022
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A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials
S. Goswami, M. Yin, Y. Yu, and G. E. Karniadakis · 2022
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Certified machine learning: A posteriori error estimation for physics-informed neural networks
B. Hillebrecht and B. Unger · 2022
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Error estimates for DeepONets: A deep learning framework in infinite dimensions
S. Lanthaler, S. Mishra, and G. E. Karniadakis · 2022
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A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data
L. Lu, X. Meng, S. Cai, Z. Mao, S. Goswami, Z. Zhang, and G. E. Karniadakis · 2022
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Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs
S. Mishra and R. Molinaro · 2022
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J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, p. Hassanzadeh, K. Kashinath, and A. Anandkumar · 2022
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