  This paper is concerned with the approximation of the solution of partial
differential equations by means of artificial neural networks. Here a
feedforward neural network is used to approximate the solution of the partial
differential equation. The learning problem is formulated as a least squares
problem, choosing the residual of the partial differential equation as a loss
function, whereas a multilevel Levenberg-Marquardt method is employed as a
training method. This setting allows us to get further insight into the
potential of multilevel methods. Indeed, when the least squares problem arises
from the training of artificial neural networks, the variables subject to
optimization are not related by any geometrical constraints and the standard
interpolation and restriction operators cannot be employed any longer. A
heuristic, inspired by algebraic multigrid methods, is then proposed to
construct the multilevel transfer operators. Numerical experiments show
encouraging results related to the efficiency of the new multilevel
optimization method for the training of artificial neural networks, compared to
the standard corresponding one-level procedure.
