Phase-based spatial ordinal patterns for characterizing oscillatory dynamics

What the math gives to ML

The paper provides a constructive symbolic coarse-graining of spatial oscillator states: overlapping triples of local phases are mapped to 13 weak-order patterns, including patterns representing near-synchrony, and their empirical entropy tracks spatial organization and transient regime changes. This is transferable to recurrent, state-space, graph, and complex-valued neural networks whose hidden units have an interpretable spatial or graph ordering. The most promising implementation is a phase-organization monitor and auxiliary regularizer that detects hidden-state synchronization collapse or encourages a prescribed mixture of coherent clusters and phase gradients. Its key falsifiable signature is that networks with identical global phase coherence can have significantly different local ordinal-pattern distributions and entropy, while globally synchronized states should produce a sharp entropy decrease.