Geodesic
Training a neural network for a hyperbolic equation by using a quasiclassical functional
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 3, Tome 237 (2024), pp. 76-86

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We study the problem of constructing a loss functional based on the quasiclassical variational principle for training a neural network, which approximates solutions of a hyperbolic equation. Using the method of symmetrizing operator proposed by V. M. Shalov, for the second-order hyperbolic equation, we construct a variational functional of the boundary-value problem, which involves integrals over the domain of the boundary-value problem and a segment of the boundary, depending on first-order derivatives of the unknown function. We demonstrate that the neural network approximating the solution of the boundary-value problem considered can be trained by using the constructed variational functional.
Keywords: variational principle, hyperbolic equation, neural network, loss functional 
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     author = {S. G. Shorokhov},
     title = {Training a neural network for a hyperbolic equation by using a quasiclassical functional},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {76--86},
     publisher = {mathdoc},
     volume = {237},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2024_237_a5/}
}
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S. G. Shorokhov. Training a neural network for a hyperbolic equation by using a quasiclassical functional. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 3, Tome 237 (2024), pp. 76-86. http://geodesic.mathdoc.fr/item/INTO_2024_237_a5/