Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators
This work addresses the challenge of improving accuracy in learning dynamical systems from noisy data, which is incremental as it builds on existing numerical integration techniques.
The paper tackles the problem of training neural networks to approximate vector fields of dynamical systems from noisy data by introducing the mean inverse integrator (MII) method, which averages multiple trajectories from numerical integrators like Runge-Kutta methods. The result is a significantly lower error compared to using the integrator alone, demonstrated with experiments on chaotic Hamiltonian systems.
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.