Deep projection networks for learning time-homogeneous dynamical systems

We consider the general class of time-homogeneous dynamical systems, both discrete and continuous, and study the problem of learning a meaningful representation of the state from observed data. This is instrumental for the task of learning a forward transfer operator of the system, that in turn can be used for forecasting future states or observables. The representation, typically parametrized via a neural network, is associated with a projection operator and is learned by optimizing an objective function akin to that of canonical correlation analysis (CCA). However, unlike CCA, our objective avoids matrix inversions and therefore is generally more stable and applicable to challenging scenarios. Our objective is a tight relaxation of CCA and we further enhance it by proposing two regularization schemes, one encouraging the orthogonality of the components of the representation while the other exploiting Chapman-Kolmogorov's equation. We apply our method to challenging discrete dynamical systems, discussing improvements over previous methods, as well as to continuous dynamical systems.

Estimating Koopman operators with sketching to provably learn large scale dynamical systems

The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems. Estimators such as principal component regression (PCR) or reduced rank regression (RRR) in kernel spaces can be shown to provably learn Koopman operators from finite empirical observations of the system's time evolution. Scaling these approaches to very long trajectories is a challenge and requires introducing suitable approximations to make computations feasible. In this paper, we boost the efficiency of different kernel-based Koopman operator estimators using random projections (sketching). We derive, implement and test the new "sketched" estimators with extensive experiments on synthetic and large-scale molecular dynamics datasets. Further, we establish non asymptotic error bounds giving a sharp characterization of the trade-offs between statistical learning rates and computational efficiency. Our empirical and theoretical analysis shows that the proposed estimators provide a sound and efficient way to learn large scale dynamical systems. In particular our experiments indicate that the proposed estimators retain the same accuracy of PCR or RRR, while being much faster.