Resolving Emergent Beat Patterns Through Hybrid Bayesian Learning of Multilayered Stochastic Hierarchical Delay Models
Modeling emergent, multiscale patterns in complex systems remains a persistent interdisciplinary challenge, particularly when deciphering transient, highly coupled dynamics from severely limited data. To overcome the inherent spectral sparsity of standard finite-dimensional stochastic models, we introduce Multilayered Stochastic Hierarchical Delay Models (MSHDMs). By embedding dynamics within a hierarchical delay structure, MSHDMs leverage an infinite-dimensional phase space to engineer the high spectral density required for generating complex Amplitude-Frequency Modulation (AM-FM) dynamics, avoiding data-heavy neural network parameterizations. To reliably calibrate these sensitive structures from short observational records, we deploy a hybrid offline-online Bayesian optimization framework. Bypassing the failure points of traditional trajectory matching, our algorithm autonomously learns optimal latent coordinates and continuous fractional delays by strictly enforcing spectral consistency against the empirical Global Wavelet Spectrum. Applying this methodology to high-resolution satellite observations of continental cloud fields, the resulting stochastic emulator captures the full spatiotemporal coherence of the turbulent system using just 6 hyperparameters. The wavelet scalograms demonstrate how the model natively generates emergent wave-packet dynamics and cross-scale energy cascades, recovering semidiurnal, mesoscale, and individual cloud timescales. Supported by rigorous mathematical foundations and robust data-driven calibrations, MSHDMs thus provide a highly compressible, general-purpose tool for resolving latent AM-FM beat patterns across diverse disciplines.