Graph-Recurrent Learning of Emergent Collective Dynamics

Ali Rayat; Acacia Lee; Gia-Wei Chern

cond-mat.str-elOct 22, 2026

Graph-Recurrent Learning of Emergent Collective Dynamics

Authors: Ali Rayat, Acacia Lee, Gia-Wei Chern

Organizations: Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

Abstract

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.

Figures & tables

Figure 1: Schematic of the graph-recurrent update. The upper lattices represent successive physical configurations $\bm{\phi}(t)$ , and the lower lattices represent the corresponding site-resolved hidden states $\bm{h}(t)$ . The labels $t-1$ , $t$ , and $t+1$ denote successive prediction steps. Neighboring sites contribute to both the spatial representation of the physical field and the evolution of the hidden states. The colored arrows indicate the exchange of information between the physical and hidden states. The schematic shows the coupled evolution; the operations within one time step are given in the text.

Figure 2: Normalized temporal autocorrelation $A(\tau)$ of the CDW order parameter, comparing GRNN predictions (green) with reference adiabatic dynamics computed using KPM (magenta). The GRNN closely reproduces the decay of temporal correlations over the full interval $0\leq\tau\leq 400$ .

Figure 3: Equal-time CDW correlation function $C(R,t)$ versus scaled separation $R=x/L(t)$ at $t=40$ , $80$ , $160$ , and $320$ , obtained from KPM reference dynamics (left) and GRNN predictions (right). The GRNN captures the spatial correlation profiles, while the collapse of curves at different times in both panels demonstrates dynamical scaling.

Figure 4: Autoregressive GRNN simulation of CDW domain coarsening on a $200\times 200$ lattice at $t=0$ , $80$ , and $6400$ . Color indicates the local order parameter $\phi$ on the scale shown at right, from $-1$ to $1$ . The initially disordered field evolves into domains of the two symmetry-related charge-ordered states.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: One-step GRNN computation. Graph layers aggregate neighboring physical features into the embedding $\bm{z}_{i}(t)$ ; the graph-coupled GRU combines this embedding with local and neighboring hidden states to produce $\bm{h}_{i}^{+}\equiv\bm{h}_{i}(t+1)$ . The readout MLP uses $\bm{z}_{i}(t)$ and the updated memory to predict $\bm{\delta}_{i}(t)$ , which enters the physical update map $\mathcal{R}_{\phi}$ . The upper bypass carries the present physical field directly to this map. Spatial and recurrent aggregation are indicated by orange and violet paths, respectively. All learned parameters are shared across lattice sites.

Quantity	Value
Hopping $t_{\rm nn}$ ; coupling $g$	$-1$ ; $3.5$
Elastic constants $k$ , $\kappa$ ; mass $m$	$1$ , $0.18$ ; $5$
Filling; temperature $k_{\rm B}T$	$\overline{n}=1/2$ ; $0.3$
Reference lattice	$40\times 40$ sites
Integration step $\delta t$ ; integrator damping setting	$0.04$ ; $0.2$
Steps per trajectory; final time	$10^{4}$ ; $400$

Appendix

Table 1: Model and reference-trajectory settings. The damping value refers to the integrator setting; no numerical value for

\gamma

in Eq. ( 10 ) is inferred from it.

Figure 6: Microscopic lattice pattern and emergent Ising-like CDW field in a representative $40\times 40$ configuration. Left: displacement $Q_{i}$ , whose alternating pattern tracks the local charge modulation through electron–lattice coupling. Right: staggered neighbor-density contrast $\phi_{i}=(-1)^{x_{i}+y_{i}}(n_{i}-\frac{1}{4}\sum_{j\in\mathcal{N}_{i}}n_{j})$ evaluated for the same configuration. The staggered factor removes the checkerboard oscillation, so red and blue regions mark the two symmetry-related CDW domains; intermediate values occur near domain walls. The color bars show the values plotted in each panel.

Figure 7: Time-dependent structure factor $S(\bm{q},t)$ of the staggered CDW field on a $40\times 40$ lattice. The upper row shows KPM reference trajectories and the lower row shows autoregressive GRNN predictions at $t=0$ , $40$ , and $400$ (left to right). Momentum is measured relative to the checkerboard ordering wave vector, so the ordering peak of the staggered field is centered at $\bm{q}=\bm{0}$ . Its narrowing as time increases reflects growing CDW domains; the GRNN reproduces the corresponding momentum-space pattern. Dashed squares outline the first Brillouin zone.

Component or procedure	Setting
Observable channels	$M=1$
Graph layers and channel sequence	$3$ ; $1\to 128\to 96\to 64$
Graph activation and aggregation	GELU; mean over nearest neighbors
Recurrent unit	GRU with local and neighbor hidden inputs
Readout	MLP with four hidden layers, GELU
Bounded residual parameters	$\epsilon=10^{-5}$ ; $\alpha$ initialized at $0.1$

Appendix

Table 2: GRNN architecture and training settings used for the Holstein demonstration.

Figure 8: Ablation study of the graph-recurrent neural network architecture. Temporal autocorrelation functions

A(\tau)

predicted by five candidate models (

C_{1}

–

C_{5}

) are compared with the KPM reference. The candidates differ in their use of recurrent memory and spatially aggregated hidden states, as summarized in Table 3 . The full GRNN architecture (

C_{5}

) and the model with spatially coupled recurrent gates (

C_{3}

) exhibit the closest agreement with the reference dynamics.

Model	GRU	$\overline{\bm{h}}_{i}$ in gates	$\overline{\bm{h}}_{i}$ in candidate	RMSE
$C_{1}$	No	–	–	0.032842
$C_{2}$	Yes	No	No	0.018433
$C_{3}$	Yes	Yes	No	0.006224
$C_{4}$	Yes	No	Yes	0.017147
$C_{5}$	Yes	Yes	Yes	0.006148

Appendix

Table 3: Comparison of the candidate architectures considered in the ablation study. All models use the same GNN backbone with feature dimensions

1\rightarrow 128\rightarrow 96\rightarrow 64

. Recurrent models employ a hidden-state dimension of 96. The RMSE quantifies the discrepancy between the predicted temporal autocorrelation function and the KPM reference.

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Mar 2, 2026J/K move · Enter open · S save

Graph neural network force fields for adiabatic dynamics of lattice Hamiltonians

Yunhao Fan, Gia-Wei Chern

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Graph Neural Networks Hamiltonian Dynamics Dynamical Systems Equivariant GNNs

Open paper AskPDF arXiv

Scalable and symmetry-consistent force-field models are essential for extending quantum-accurate simulations to large spatiotemporal scales. While descriptor-based neural networks can incorporate lattice symmetries through carefully engineered features, we show that graph neural networks (GNNs) provide a conceptually simpler and more unified alternative in which discrete lattice translation and point-group symmetries are enforced directly through local message passing and weight sharing. We develop a GNN-based force-field framework for the adiabatic dynamics of lattice Hamiltonians and demonstrate it for the semiclassical Holstein model. Trained on exact-diagonalization data, the GNN achieves high force accuracy, strict linear scaling with system size, and direct transferability to large lattices. Enabled by this scalability, we perform large-scale Langevin simulations of charge-density-wave ordering following thermal quenches, revealing dynamical scaling and anomalously slow sub--Allen--Cahn coarsening. These results establish GNNs as an elegant and efficient architecture for symmetry-aware, large-scale dynamical simulations of correlated lattice systems.

Abstract

Recurrent GNNs

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