Classifying Directional Trajectories Near Criticality in the Three-State Majority-Vote Model with Deep Belief Networks and Bidirectional GRUs

In this work, we investigate whether the latent representations learned by a Deep Belief Network (DBN) and a Bidirectional Gated Recurrent Unit (Bi-GRU) can discriminate among four dynamically distinct trajectory types in the three-state majority vote model (MV3): approach from disorder, approach from order, departure to disorder, and departure to order. The DBN, pre-trained in an unsupervised manner on static equilibrium samples via a Gaussian-Bernoulli Restricted Boltzmann Machine input layer and architecture $784 \to 4096 \to 225 \to 81$, encodes each lattice snapshot into an 81-dimensional latent vector. A t-SNE analysis of the DBN latent space reveals only partial separation of the four trajectory types, reflecting the fact that a model trained on static configurations cannot fully resolve directional temporal structure. A two-layer Bi-GRU classifier, trained on sequences of DBN-encoded snapshots of length $T = 50$, achieves near-perfect separation of all four trajectory types in its hidden state space, as confirmed by t-SNE visualization on both training and test sets. Furthermore, a sliding-window application of the trained Bi-GRU to continuous MV3 dynamics demonstrates its ability to sense the system's current dynamical regime in real-time. These results establish a principled hierarchical architecture for detecting and classifying critical transitions in agent-based opinion dynamics models.

Ptolemy's Equant Equates to a Universal Dynamical Clock via Machine Learning

Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem. Here we establish the principle of a universal dynamical clock, a physical perspective in which oscillations of arbitrary dimensionality and geometry are equivalently represented as uniform rotation through an equant-induced nonlinear viewing coordinate, inspired by Ptolemy's equant and formalised through an areal-uniformity principle reminiscent of Kepler's second law. Using a machine-learning framework, we demonstrate the existence of such an equant for a broad class of oscillatory dynamics and construct the associated dynamical clock and phase dynamics under additive forces, including noise, periodic perturbations, and coupling. Its value in uncovering new physical rules and phenomena is demonstrated by four findings: (i) collective oscillations in Escherichia coli populations obey a previously unexplained superlinear scaling law, resolving a long-standing open problem posed in 2004; (ii) the response mechanisms of engineered genetic circuits to changes in gene expression and environmental conditions; (iii) a classical-mechanics counterpart of the Berry geometric phase emerges naturally from the phase of the dynamical clock; and (iv) optimal equant non-uniformity provides a geometric early-warning signal for critical transitions and enables prediction of critical parameters. By providing operational and system-agnostic phase dynamics that can be constructed directly from data, the dynamical clock enables principled classification, comparison, and control of oscillatory systems, and offers a new route to understanding how specific dynamical regimes support distinct functional behaviours in networked systems.