  We introduce the mean inverse integrator (MII), a novel approach to increase
the accuracy when training neural networks to approximate vector fields of
dynamical systems from noisy data. This method can be used to average multiple
trajectories obtained by numerical integrators such as Runge-Kutta methods. We
show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular
advantages when used in connection with MII. When training vector field
approximations, explicit expressions for the loss functions are obtained when
inserting the training data in the MIRK formulae, unlocking symmetric and
high-order integrators that would otherwise be implicit for initial value
problems. The combined approach of applying MIRK within MII yields a
significantly lower error compared to the plain use of the numerical integrator
without averaging the trajectories. This is demonstrated with experiments using
data from several (chaotic) Hamiltonian systems. Additionally, we perform a
sensitivity analysis of the loss functions under normally distributed
perturbations, supporting the favorable performance of MII.
