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

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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. 
@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},
     publisher = {mathdoc},
     volume = {499},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_499_a6/}
}
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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/