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The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs.
Symmetric positive linear differential equations
K. O. Friedrichs · 1958
Earlier work this paper cites.
Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
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Kurt Hornik, Maxwell Stinchcombe, and Halbert White · 1989
Earlier work this paper cites.
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Hyuk Lee and In Seok Kang · 1990
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Universal approximation bounds for superpositions of a sigmoidal function
A. R. Barron · 1993
Earlier work this paper cites.
Neural-network-based approximations for solving partial differential equations
M. W. M. G. Dissanayake and N. Phan-Thien · 1994
Earlier work this paper cites.
Analog cellular neural network with application to partial differential equations with variable mesh-size
D. Gobovic and M. E. Zaghloul · 1994
Earlier work this paper cites.
Artificial neural networks for solving ordinary and partial differential equations
I. E. Lagaris, A. Likas, and D. I. Fotiadis · 1998
Earlier work this paper cites.
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Earlier work this paper cites.
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Earlier work this paper cites.
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Earlier work this paper cites.
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H. Yserentant · 2005
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T. Kaufmann, C. Engström, and C. Fumeaux · 2010
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