Out-of-Domain Generalization in Dynamical Systems Reconstruction
Niclas Göring Florian Hess Manuel Brenner Zahra Monfared Daniel Durstewitz
In science we are interested in finding the governing equations, the
dynamical rules, underlying empirical phenomena. While traditionally scientific
models are derived through cycles of human insight and experimentation,
recently deep learning (DL) techniques have been advanced to reconstruct
dynamical systems (DS) directly from time series data. State-of-the-art
dynamical systems reconstruction (DSR) methods show promise in capturing
invariant and long-term properties of observed DS, but their ability to
generalize to unobserved domains remains an open challenge. Yet, this is a
crucial property we would expect from any viable scientific theory. In this
work, we provide a formal framework that addresses generalization in DSR. We
explain why and how out-of-domain (OOD) generalization (OODG) in DSR profoundly
differs from OODG considered elsewhere in machine learning. We introduce
mathematical notions based on topological concepts and ergodic theory to
formalize the idea of learnability of a DSR model. We formally prove that
black-box DL techniques, without adequate structural priors, generally will not
be able to learn a generalizing DSR model. We also show this empirically,
considering major classes of DSR algorithms proposed so far, and illustrate
where and why they fail to generalize across the whole phase space. Our study
provides the first comprehensive mathematical treatment of OODG in DSR, and
gives a deeper conceptual understanding of where the fundamental problems in
OODG lie and how they could possibly be addressed in practice.