Finding the Optimal Continuous Model for Discrete Data by Neural Network Interpolation of Fractional Iteration

Given the complete knowledge of the state variables of a dynamicalsystem at fixed intervals, it is possible to construct a mapping, which is a perfectdiscrete time model of the system. To embed this into a continuum, the translationequation has to be solved for this mapping. However, in general, neitherexistence nor uniqueness of solutions can be guaranteed, but fractional iteratesof the mapping computed by a neural network can provide regularized solutionsthat exactly comply with the laws of physics for several examples. Here weextend this method to continuous embeddings which represent the true trajectoriesof the dynamical system