“A key enabling assumption, sometimes called the manifold hypothesis [14], is that the data lie on or near a low-dimensional manifold; for physical systems with dissipation, such manifolds can often be rigorously shown to exist [15–18]. These manifolds enable a low-dimensional latent state representation, and hence, low-dimensional dynamical models. Linear manifold learning techniques, such as principal component analysis, cannot learn the nonlinear manifolds that represent most systems in nature. To do so, we require nonlinear methods, some of which are developed in [19–25] and reviewed in [26].“
Floryan, Daniel, and Michael D. Graham. “Data-driven discovery of intrinsic dynamics.” Nature Machine Intelligence 4.12 (2022): 1113-1120.
From <https://www.nature.com/articles/s42256-022-00575-4>
GC22A-04 Can ML beats chaos?
Abstract
“Chaos is typically blamed for the lack of predictability beyond a forecasting time window, which is on the order of 10 days for weather forecasting. However, on the one hand, most turbulent and chaotic systems exhibit strong coherence in the flow, such as synoptic events in weather or coherent structures in turbulence. On the other hand, most physical model might have additional structural errors that limit their capacity to correctly forecast beyond a certain time horizon, independent of chaos.
We will show in this presentation that most chaotic and turbulent flows can be predicted on relatively long range, at least longer than expected with both physical models and standard deep learning, using a combination of a reduced order model (that captures the low-dimensional coherent structures in the flow) and generative AI to obtain the crisp results and details of the flow. We will conclude stating that current AI-based weather models might not have achieved a plateau in performance yet, especially at longer time scales, and that physics-based weather model still have room for improvements. Reduced order models might not be able to beat chaos but can lead to much longer-range prediction than currently expected.“
https://agu.confex.com/agu/agu24/meetingapp.cgi/Paper/1522150