Keywords: Deep neural networks, Hamiltonian systems, ODE discretization, Runge-Kutta, Geometric numerical integration
TL;DR: The mono-implicit Runge-Kutta methods are explicit for inverse ODE problems and could be used in the novel mean inverse integrator to approximate ODE vector fields from noisy data with improved accuracy, compared to one-step methods.
Abstract: Decades of research have been spent on classifying the properties of numerical integrators when solving ordinary differential equations (ODEs). Here, a first step is taken to examine the properties of numerical integrators when used to learn dynamical systems from noisy data with neural networks. Mono-implicit Runge--Kutta (MIRK) methods are a class of integrators that can be considered explicit for inverse problems. The symplectic property is useful when learning the dynamics of Hamiltonian systems. Unfortunately, a proof shows that symplectic MIRK methods have a maximum order of $p=2$. By taking advantage of the inverse explicit property, a novel integration method called the mean inverse integrator, tailored for solving inverse problems with noisy data, is introduced. As verified in numerical experiments on different dynamical systems, this method is less sensitive to noise in the data.