Certified machine learning: A posteriori error estimation for physics-informed neural networks

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , ser. Applied mathematical sciences. New York, NY, USA: Springer, 1983

1983

Earlier work this paper cites.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks , vol. 2, no. 5, pp. 359–366, 1989

1989

Earlier work this paper cites.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. , vol. 45, pp. 503–528, 1989

1989

Earlier work this paper cites.

C. Anderson, “Learning to control an inverted pendulum using neural networks,” IEEE Control Syst. Mag. , vol. 9, no. 3, pp. 31–37, 1989

1989

Earlier work this paper cites.

K. J. Hunt, D. Sbarbaro, R. Żbikowski, and P. J. Gawthrop, “Neural networks for control systems—a survey,” Automatica J. IFAC , vol. 28, no. 6, pp. 1083–1112, 1992

1992

Earlier work this paper cites.

M. Leshno, V. Ya. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforward networks with a nonpolynomial activation function can approximate any function,” Neural Networks , vol. 6, no. 6, pp. 861–867, 1993

1993

Earlier work this paper cites.

A. R. Barron, “Approximation and estimation bounds for artificial neural networks,” Mach Learn , vol. 14, pp. 115–133, 1994

1994

Earlier work this paper cites.

R. Verfürth, “A posteriori error estimates for nonlinear problems,” Math. Comp. , 1994

1994

Earlier work this paper cites.

W. Gautschi, Numerical analysis . Springer Science & Business Media, 1997

1997

Earlier work this paper cites.

A. Pinkus, “Approximation theory of the MLP model in neural networks,” Acta Numer. , vol. 8, pp. 143–195, 1999

1999

Cited alongside, same era.

F. Cao, T. Xie, and Z. Xu, “The estimate for approximation error of neural networks: A constructive approach,” Neurocomputing , vol. 71, pp. 626–630, 2008

2008

Cited alongside, same era.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I , 3rd ed. Berlin, Heidelberg: Springer, 2008

2008

Cited alongside, same era.

C. Ramirez, R. Sanchez, V. Kreinovich, and M. Argaez, “ \sqrt{} (x2 + μ \mu ) is the most computationally efficient smooth approximation to —x—: a proof,” Journal of Uncertain Systems , vol. 8, 2014

2014

Cited alongside, same era.

J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations , ser. SpringerBriefs in Mathematics. Cham, Switzerland: Springer, 2016

I. Cortés-Ciriano and A. Bender, “Reliable prediction errors for deep neural networks using test-time dropout,” J. Chem. Inf. Model. , vol. 59, no. 7, pp. 3330–3339, jun 2019

2019

Later among the works it cites.

A. D. Jagtap, K. Kawaguchi, and G. E. Karniadakis, “Adaptive activation functions accelerate convergence in deep and physics-informed neural networks,” J. Comput. Phys. , vol. 404, p. 109136, 2020

2020

Later among the works it cites.

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nature Reviews Physics , vol. 3, pp. 422–440, 2021

2021

Later among the works it cites.

R. Leiteritz, P. Buchfink, B. Haasdonk, and D. Pflüger, “Surrogate-data-enriched physics-aware neural networks,” ArXiv e-print 2112.05489 , 2021. [Online]. Available: https://arxiv.org/abs/2112.05489

Original

2021

Later among the works it cites.