Geodesic
A posteriori error control of approximate solutions to boundary value problems constructed by neural networks
Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part I, Tome 499 (2021), pp. 77-104 
A. V. Muzalevsky; S. I. Repin. A posteriori error control of approximate solutions to boundary value problems constructed by neural networks. Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part I, Tome 499 (2021), pp. 77-104. http://geodesic.mathdoc.fr/item/ZNSL_2021_499_a6/
@article{ZNSL_2021_499_a6,
     author = {A. V. Muzalevsky and S. I. Repin},
     title = {A posteriori error control of approximate solutions to boundary value problems constructed by neural networks},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {77--104},
     year = {2021},
     volume = {499},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_499_a6/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper discusses how to verify the quality of approximate solutions to partial differential equations constructed by deep neural networks. A posterior error estimates of the functional type, that have been developed for a wide range of boundary value problems, are used to solve this problem. It is shown, that they allow one to construct guaranteed two-sided estimates of global errors and get distribution of local errors the error over the domain. The corresponding results of numerical experiments are presented for a boundary value problem of an elliptic type. They show that the estimates provide much more reliable information than the so-called loss function, which is commonly used as a quality criterion training neural network models.

[1] W. E, J. Han, A. Jentzen, “Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations”, Commun. Math. Stat., 5 (2017), 349–380 | DOI | MR | Zbl

[2] J. He, L. Li, J. Xu, C. Zheng, ReLU deep neural networks and linear finite elements, arXiv: 1807.03973v2

[3] R. Khudorozhkov, S. Tsimfer, A. Koryagin, PyDEns framework for solving differential equations with deep learning, 2019, arXiv: 1909.11544 [cs.LG]

[4] I. E. Lagaris, A. Likas, D. I. Fotiadis, “Artificial neural networks for solving ordinary and partial differential equations”, IEEE Transactions on Neural Networks, 9:5, 987–1000 | DOI

[5] W. E, B. Yu, “The Deep Ritz Method: A deep learning-based numerical algorithm for solving variational problems”, Commun. Math. Stat., 6 (2018), 1–12 | MR

[6] O. Pironneau, “Parameter identification of a fluid-structure system by deep-learning with an Eulerian formulation”, Methods Appl. Anal., 26:3 (2019), 281–290 | DOI | MR | Zbl

[7] F. Regazzonia, L. Dede, A. Quarteroni, “Machine learning for fast and reliable solution of time-dependent differential equations”, J. Comput. Phys., 397 (1088), 108852 | DOI

[8] E. Samaniego, C. Anitescu, S. Goswami, V.M. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, T. Rabczuka, “An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications”, Comput. Methods Appl. Mech. Engrg., 362 (2020), 112790 | DOI | MR | Zbl

[9] S. Repin, A posteriori estimates for partial differential equations, Walter de Gruyter GmbH Co. KG, Berlin, 2008

[10] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Stuttgart, 1996 | Zbl

[11] S. Repin, “A posteriori error estimation for nonlinear variational problems by duality theory”, Zap. Nauchn. Semin., 243, 1997, 201–214 | Zbl

[12] S. Repin, “A posteriori error estimation for variational problems with uniformly convex functionals”, Math. Comp., 69 (2000), 481–500 | DOI | MR | Zbl

[13] C. I. Repin, “Dvustoronnie otsenki otkloneniya ot tochnogo resheniya dlya ravnomerno ellipticheskikh uravnenii”, Trudy S.-Peterburgskogo matematicheskogo obschestva, 9 (2001), 148–179 | Zbl

[14] S. Repin, “Estimates of deviation from exact solutions of initial-boundary value problems for the heat equation”, Rend. Mat. Acc. Lincei, 13 (2002), 121–133 | Zbl

[15] S. Repin, S. Sauter, A. Smolianski, “Two-sided a posteriori error estimates for mixed formulations of elliptic problems”, SIAM J. Num. Analysis, 45 (2007), 928–945 | DOI | Zbl

[16] S. I. Repin, M. E. Frolov, “Aposteriornye otsenki pogreshnosti priblizhennykh reshenii kraevykh zadach ellipticheskogo tipa”, ZhVMiMF, 42:12 (2002), 1704–1716 | MR | Zbl

[17] O. Mali, P. Nettaanmäki, S. Repin, Accuracy verification methods. Theory and Algorithms, Springer, Berlin, 2014 | Zbl

[18] P. A. Raviart, J. M. Thomas, “A mixed finite element method for 2-nd order elliptic problems”, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606, eds. Galligani I., Magenes E., Springer, Berlin–Heidelberg, 1977

[19] J. Sirignano, K. Spiliopoulos, “DGM: A deep learning algorithm for solving partial differential equations”, J. Comput. Phys., 375 (2018), 1339–1364 | DOI | MR | Zbl

[20] H. Gomez, L. Lorenzis, “The variational collocation method”, Computer Methods in Applied Mechanics and Engineering, 309, 2016, 152–181 | DOI | Zbl