  Given (small amounts of) time-series' data from a high-dimensional,
fine-grained, multiscale dynamical system, we propose a generative framework
for learning an effective, lower-dimensional, coarse-grained dynamical model
that is predictive of the fine-grained system's long-term evolution but also of
its behavior under different initial conditions. We target fine-grained models
as they arise in physical applications (e.g. molecular dynamics, agent-based
models), the dynamics of which are strongly non-stationary but their transition
to equilibrium is governed by unknown slow processes which are largely
inaccessible by brute-force simulations. Approaches based on domain knowledge
heavily rely on physical insight in identifying temporally slow features and
fail to enforce the long-term stability of the learned dynamics. On the other
hand, purely statistical frameworks lack interpretability and rely on large
amounts of expensive simulation data (long and multiple trajectories) as they
cannot infuse domain knowledge. The generative framework proposed achieves the
aforementioned desiderata by employing a flexible prior on the complex plane
for the latent, slow processes, and an intermediate layer of physics-motivated
latent variables that reduces reliance on data and imbues inductive bias. In
contrast to existing schemes, it does not require the a priori definition of
projection operators from the fine-grained description and addresses
simultaneously the tasks of dimensionality reduction and model estimation. We
demonstrate its efficacy and accuracy in multiscale physical systems of
particle dynamics where probabilistic, long-term predictions of phenomena not
contained in the training data are produced.
