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A classic approach for solving differential equations with neural networksbuilds upon neural forms, which employ the differential equation with adiscretisation of the solution domain. Making use of neural forms fortime-dependent differential equations, one can apply the recently developedmethod of domain fragmentation. That is, the domain may be split into severalsubdomains, on which the optimisation problem is solved. In classic adaptivenumerical methods, the mesh as well as the domain may be refined or decomposed,respectively, in order to improve accuracy. Also the degree of approximationaccuracy may be adapted. It would be desirable to transfer such important andsuccessful strategies to the field of neural network based solutions. In thepresent work, we propose a novel adaptive neural approach to meet this aim forsolving time-dependent problems. To this end, each subdomain is reduced in sizeuntil the optimisation is resolved up to a predefined training accuracy. Inaddition, while the neural networks employed are by default small, we propose ameans to adjust also the number of neurons in an adaptive way. We introduceconditions to automatically confirm the solution reliability and optimisecomputational parameters whenever it is necessary. Results are provided forseveral initial value problems that illustrate important computationalproperties of the method alongside. In total, our approach not only allows toanalyse in high detail the relation between network error and numericalaccuracy. The new approach also allows reliable neural network solutions overlarge computational domains.

Adaptive neural domain refinement for solving time-dependent differential equations