  We present a new approach to using neural networks to approximate the
solutions of variational equations, based on the adaptive construction of a
sequence of finite-dimensional subspaces whose basis functions are realizations
of a sequence of neural networks. The finite-dimensional subspaces are then
used to define a standard Galerkin approximation of the variational equation.
This approach enjoys a number of advantages, including: the sequential nature
of the algorithm offers a systematic approach to enhancing the accuracy of a
given approximation; the sequential enhancements provide a useful indicator for
the error that can be used as a criterion for terminating the sequential
updates; the basic approach is largely oblivious to the nature of the partial
differential equation under consideration; and, some basic theoretical results
are presented regarding the convergence (or otherwise) of the method which are
used to formulate basic guidelines for applying the method.
