  Several neural network approaches for solving differential equations employ
trial solutions with a feedforward neural network. There are different means to
incorporate the trial solution in the construction, for instance one may
include them directly in the cost function. Used within the corresponding
neural network, the trial solutions define the so-called neural form. Such
neural forms represent general, flexible tools by which one may solve various
differential equations. In this article we consider time-dependent initial
value problems, which require to set up the neural form framework adequately.
The neural forms presented up to now in the literature for such a setting can
be considered as first order polynomials. In this work we propose to extend the
polynomial order of the neural forms. The novel collocation-type construction
includes several feedforward neural networks, one for each order. Additionally,
we propose the fragmentation of the computational domain into subdomains. The
neural forms are solved on each subdomain, whereas the interfacing grid points
overlap in order to provide initial values over the whole fragmentation. We
illustrate in experiments that the combination of collocation neural forms of
higher order and the domain fragmentation allows to solve initial value
problems over large domains with high accuracy and reliability.
