Data-driven modelling of autonomous and forced dynamical systems

Abstract

The paper demonstrates that invariant foliations are accurate, data-efficient and practical tools for data-driven modelling of physical systems. Invariant foliations can be fitted to data that either fill the phase space or cluster about an invariant manifold. Invariant foliations can be fitted to a single trajectory or multiple trajectories. Over and underfitting are eliminated by appropriately choosing a function representation and its hyperparameters, such as polynomial orders. The paper extends invariant foliations to forced and parameter dependent systems. It is assumed that forcing is provided by a volume preserving map, and therefore the forcing can be periodic, quasi-periodic or even chaotic. The method utilises full trajectories, hence it is able to predict long-term dynamics accurately. We take into account if a forced system is reducible to an autonomous system about a steady state, similar to how Floquet theory guarantees reducibility for periodically forced systems. In order to find an invariant manifold, multiple invariant foliations are calculated in the neighbourhood of the invariant manifold. Some of the invariant foliations can be linear, while others nonlinear but only defined in a small neighbourhood of an invariant manifold, which reduces the number of parameters to be identified. An invariant manifold is recovered as the zero level set of one or more of the foliations. To interpret the results, the identified mathematical models are transformed to a canonical form and instantaneous frequency and damping information are calculated.