Numerical integrators for learning dynamical systems from noisy data

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.