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Collective ordering in correlated electron systems produces emergent, non-Markovian dynamics that cannot be closed at the level of coarse-grained observables alone. This paper proposes a graph-recurrent neural network (GRNN) that fuses gated recurrent memory with graph message passing to learn finite-time evolution maps directly from trajectories, bypassing repeated microscopic force evaluations. Site-resolved hidden states encode unresolved history and communicate across neighboring sites via graph-coupled hidden messages, giving a scalable, translationally invariant update rule. The authors demonstrate the approach on charge-density-wave domain coarsening in the semiclassical Holstein model, showing quantitative agreement with kernel-polynomial-method reference dynamics.
Collective ordering in correlated electron systems emerges from microscopic dynamics, yet the evolution of observable textures need not admit a closed, memoryless description. We introduce a graph-recurrent neural network (GRNN) framework that learns such emergent dynamics directly from trajectories without specifying microscopic equations of motion. The framework integrates gated recurrent memory with graph message passing: local hidden states encode information beyond the instantaneous observables, while spatial coupling allows this information to propagate between sites. The observable and hidden fields evolve jointly through a shared local rule, providing a scalable representation of dynamics with spatial interactions and temporal memory. We demonstrate this framework on the emergent dynamics of charge-density-wave order in the semiclassical Holstein model, successfully reproducing temporal and spatial correlations and dynamical scaling during domain coarsening. By learning finite-time evolution directly, the GRNN offers a route to efficient, data-driven descriptions of collective dynamics with unresolved microscopic degrees of freedom.