  A classic approach for solving differential equations with neural networks
builds upon neural forms, which employ the differential equation with a
discretisation of the solution domain. Making use of neural forms for
time-dependent differential equations, one can apply the recently developed
method of domain fragmentation. That is, the domain may be split into several
subdomains, on which the optimisation problem is solved. In classic adaptive
numerical methods, the mesh as well as the domain may be refined or decomposed,
respectively, in order to improve accuracy. Also the degree of approximation
accuracy may be adapted. It would be desirable to transfer such important and
successful strategies to the field of neural network based solutions. In the
present work, we propose a novel adaptive neural approach to meet this aim for
solving time-dependent problems. To this end, each subdomain is reduced in size
until the optimisation is resolved up to a predefined training accuracy. In
addition, while the neural networks employed are by default small, we propose a
means to adjust also the number of neurons in an adaptive way. We introduce
conditions to automatically confirm the solution reliability and optimise
computational parameters whenever it is necessary. Results are provided for
several initial value problems that illustrate important computational
properties of the method alongside. In total, our approach not only allows to
analyse in high detail the relation between network error and numerical
accuracy. The new approach also allows reliable neural network solutions over
large computational domains.
