Numerical and statistical analysis of NeuralODE with Runge-Kutta time integration

NeuralODE is one example for generative machine learning based on the push forward of a simple source measure with a bijective mapping, which in the case of NeuralODE is given by the flow of a ordinary differential equation. Using Liouville's formula, the log-density of the push forward measure is easy to compute and thus NeuralODE can be trained based on the maximum Likelihood method such that the Kulback-Leibler divergence between the push forward through the flow map and the target measure generating the data becomes small. In this work, we give a detailed account on the consistency of Maximum Likelihood based empirical risk minimization for a generic class of target measures. In contrast to prior work, we do not only consider the statistical learning theory, but also give a detailed numerical analysis of the NeuralODE algorithm based on the 2nd order Runge-Kutta (RK) time integration. Using the universal approximation theory for deep ReQU networks, the stability and convergence rated for the RK scheme as well as metric entropy and concentration inequalities, we are able to prove that NeuralODE is a probably approximately correct (PAC) learning algorithm.