Can a Neural ODE learn a Chaotic System?

Learning chaotic dynamical systems from data presents significant challenges due to their inherent unpredictability and high sensitivity to initial conditions. Conventional metrics of model performance, such as generalization error, often fail to capture a neural network's ability to reproduce the invariant statistics of these complex systems. In this thesis, we demonstrate through a comprehensive set of examples that pointwise accuracy (measured through generalization error) does not necessarily translate into statistical fidelity. Our evaluation leverages concepts from ergodic theory to provide a more nuanced assessment of model performance. Then, we propose and implement modifications to the training scheme by incorporating Jacobian information. We show that this approach enables the reproduction of correct physical measures for chaotic systems, which we term "statistically-accurate learning.'' We also report the failure mode of our proposed training scheme and give a theoretical explanation with shadowing lemma. Our work offers valuable insights into the limitations of traditional machine learning theory when applied to complex systems. These findings have implications for improving the reliability and interpretability of machine learning models in complex dynamical contexts, with potential applications in fields such as climate modeling, fluid dynamics, and nonlinear control systems.

Date

2024-07-28

Resource Type

Text

Resource Subtype

Thesis

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