Double projection for reconstructing dynamical systems: between stochastic and deterministic regimes

Abstract Learning stochastic models of dynamical systems from observed data is of interest in many scientific fields. For reliable use in scientific applications, trained models must be able to reproduce the observed dynamics, remain amenable to analysis by mathematical and computational tools, and their dependence on key training hyperparameters must be well understood. Here, we propose a new method for this task within the family of dynamical variational autoencoders. The proposed double projection method estimates both the system state trajectories and the noise time series from data. This approach naturally allows us to perform multi-step system evolution and to learn models with a comparatively low-dimensional state space. We evaluate the performance of the method on six benchmark problems, including both simulated and experimental data. We then analyze how the teacher forcing interval, a key parameter of the multi-step system evolution, shapes the internal dynamics of the trained model, and we compare the resulting behavior to that of deterministic models of equivalent architecture. The results show that shorter teacher forcing intervals favor predominantly deterministic dynamics, whereas longer intervals shift the model behavior toward stronger reliance on stochastic forcing. We further show that the inferred noise can play distinct roles: in one case it recovers a genuine source of stochasticity underlying the data, while in another it compensates for dynamics that the model cannot capture. Our results may provide insight into the behavior of related methods based on teacher forcing techniques and into the interpretation of learned stochastic models more broadly.