ICLR 2025

Published as a conference paper at ICLR 2025
LEARNING CHAOS IN A LINEAR WAY
Xiaoyuan Cheng1,†
Yi He1,†
Yiming Yang1,2
Xiao Xue1
Sibo Cheng3
Daniel Giles4
Xiaohang Tang2
Yukun Hu1,*
1Dynamic Systems, University College London, United Kingdom
2Statistical Science, University College London, United Kingdom
3CEREA, ENPC and EDF R&D, Institut Polytechnique de Paris, France
4Computer Science, University College London, United Kingdom
† Equal contribution
* Corresponding author (yukun.hu@ucl.ac.uk)
ABSTRACT
Learning long-term behaviors in chaotic dynamical systems, such as turbulent
flows and climate modelling, is challenging due to their inherent instability and
unpredictability. These systems exhibit positive Lyapunov exponents, which signif-
icantly hinder accurate long-term forecasting. As a result, understanding long-term
statistical behavior is far more valuable than focusing on short-term accuracy.
While autoregressive deep sequence models have been applied to capture long-term
behavior, they often lead to exponentially increasing errors in learned dynamics.
To address this, we shift the focus from simple prediction errors to preserving an
invariant measure in dissipative chaotic systems. These systems have attractors,
where trajectories settle, and the invariant measure is the probability distribution on
attractors that remains unchanged under dynamics. Existing methods generate long
trajectories of dissipative chaotic systems by aligning invariant measures, but it is
not always possible to obtain invariant measures for arbitrary datasets. We propose
the Poincar´e Flow Neural Network (PFNN), a novel operator learning framework
designed to capture behaviors of chaotic systems without any explicit knowledge of
the invariant measure. PFNN employs an auto-encoder to map the chaotic system
to a finite-dimensional feature space, effectively linearizing the chaotic evolution.
It then learns the linear evolution operators to match the physical dynamics by ad-
dressing two ...