Training a neural network for a hyperbolic equation by using a quasiclassical functional

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.