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

Abstract

Abstract. Given the complete knowledge of the state variables of a dynamical system at fixed intervals, it is possible to construct a mapping, which is a perfect discrete time model of the system. To embed this into a continuum, the translation equation has to be solved for this mapping. However, in general, neither existence nor uniqueness of solutions can be guaranteed, but fractional iterates of the mapping computed by a neural network can provide regularized solutions that exactly comply with the laws of physics for several examples. Here we extend this method to continuous embeddings which represent the true trajectories of the dynamical system.