Parameterized neural network training for the solution of a class of stiff initial value systems

Abstract

As computational intelligence techniques become more popular in almost all scientific fields and applications nowadays, there exists an active research effort to engage them in the study of classical mathematical problems. Among these techniques, the neural networks (NN), apart from their use in classification problems, can be used to approximate the behaviour of functions and their derivatives. Towards this direction, NN solution of differential equations (DEs), in both theoretical and technical point of view, is an active scientific field for the last two decades. NN solutions for DEs, once trained, have low computational cost and can be very useful as parts of more complex algorithms where needed, as well. Among the various classes of DEs, the stiff initial value problems (IVP) reveal difficulties in their numerical treatment by classical methodologies, whereas NN solutions of stiff DEs do not seem to do so. Moreover, their continuous nature and ability to be trained to solve classes of problems makes them an interesting tool. In this study, we investigate the NN solution of Inhomogeneous Linear IVPs. We incorporate to the NN a parameter that influences problem’s stiffness and train the network for a range of this. Therefore, the trained NN solution can solve different problems than the one training for. In order to reveal the good generalization properties of the NNs solution regarding the stiffness parameter, we compare them to the solutions of standard Matlab stiff solvers. The proposed solutions perform very well, similarly and in many cases better to their competitors.