Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators

Håkon Noren, Sølve Eidnes, Elena Celledoni

Keywords: Deep neural networks, Hamiltonian systems, ODE discretization, Runge-Kutta, Geometric numerical integration

TL;DR: We introduce the Mean inverse integrator (MII), a novel approach based on mono-implicit Runge-Kutta methods to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data.

Abstract: 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 favourable performance of MII.

Submission Number: 12112