The main goal of this paper is to generalize the method introduced in [M. Raissi et al., “Physics informed deep learning. I: Data-driven solutions of nonlinear partial differential equations”, Preprint, arXiv:1711.10561] for boundary value problems, introducing new loss functionals. The authors start by introducing a loss functional such that the one of [loc. cit.] can be seen as its Monte-Carlo approximation in a particular scenario.
A bridge between the minimization of the new loss functional and the approximation for the solution of well-posed boundary value problems is established in Theorem 1. The authors present several remarks justifying the use of this result to guarantee success in the application of neural networks to minimize the correspondent Monte Carlo approximation.
In Theorem 2, for linear boundary value problems, it is shown that the new loss functional is convex. Consequently, this loss functional does not have local minima. It should be pointed out that neural networks convert convex loss functionals into non-convex problems. Moreover, Theorem 2 requires the linearity of the boundary value problem.
In Section 3 is discussed a new choice of the loss functional introducing a new parameter referred as the loss weight. The main question is how we should fix this parameter, that is, what is its optimal choice with respect to a specific error measure. To give an answer to this question, the authors introduce the concept of \(\epsilon\)-closeness which can be seen as relative error for the approximate solution. Assuming that the approximate solution is \(\epsilon\)-close, the authors construct upper bounds for the interior and boundary loss terms of the loss functional depending on the true solution of the differential problem. These upper bounds are then used to fix the loss weight parameter. As the true solution is unknown, the authors replace it by the approximate solution obtaining a normalized loss functional in Section 4. To apply the results obtained in this paper, in Section 5 is discussed the network configuration, the computation of the loss functionals and the training algorithms. In Section 6, the authors compare the proposed approach with the one introduced in [loc. cit.] considering several differential problems: Laplace and Poisson equations with Dirichlet boundary conditions; a convection-diffusion equation with Dirichlet boundary conditions in \([0,1]\).
From the numerical experiments, we conclude that the approach introduced here shows, in general, higher accuracy than the method studied in [loc. cit.].