Local causal states and discrete coherent structures

Abstract

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the local causal states, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions.

Towards Unsupervised Segmentation of Extreme Weather Events

“…classification assignments of 'cylcone', 'atmospheric river', 'background', etc. ), for which an additional layer of analysis is needed on top of the local causal states [26]. From comparison with established Lagrangian Coherent Structure results we will now show that physically meaningful coherent structures are captured by our structural segmentation, and then we outline how extreme weather events may be extracted from structural segmentation of global climate data.…”

Visualizing computation in large-scale cellular automata

“…Our method is also arguably much simpler than computational mechanics (used by Hanson and Crutchfield) which requires some reverse-engineering of the rule and the construction of a finite-state transducer to generate output symbols. Although full automation has been demonstrated, this method introduces significant overhead (Rupe and Crutchfield, 2018). On the other hand, our method is sensitive to the quality of statistical estimation of the frequency histogram (see equation ( 1)) and needs enough input examples to achieve a reasonable result -examples in Figure 5 and 6 used simulations with the width of 3000 cells, ran for 6000 timesteps to obtain reliable pattern frequency estimates.…”

DisCo: Physics-Based Unsupervised Discovery of Coherent Structures in Spatiotemporal Systems

“…This means the learned representation is directly utilizable for discovering pattern and structure in the physical observable field. In particular, coherent structure in X are identified through locally broken symmetries in S [59].…”

“…Yellow boxes are new structural signatures of elliptic LCS discovered by DisCo. Because there is a single background state, colored white, in (c), all objects with a bounding box can be assigned a semantic label of coherent structure since they satisfy the local causal state definition given in [59] as spatially localized, temporally persistent deviations from generalized spacetime symmetries (i.e. local causal state symmetries).…”