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 ϕ(t) , and the lower lattices represent the corresponding site-resolved hidden states 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(τ) 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≤τ≤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 200×200 lattice at t=0 , 80 , and 6400 . Color indicates the local order parameter ϕ 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
Figure 5: One-step GRNN computation. Graph layers aggregate neighboring physical features into the embedding zi(t) ; the graph-coupled GRU combines this embedding with local and neighboring hidden states to produce hi+≡hi(t+1) . The readout MLP uses zi(t) and the updated memory to predict δi(t) , which enters the physical update map Rϕ . 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 tnn ; coupling g
−1 ; 3.5
Elastic constants k , κ ; mass m
1 , 0.18 ; 5
Filling; temperature kBT
n=1/2 ; 0.3
Reference lattice
40×40 sites
Integration step δt ; integrator damping setting
0.04 ; 0.2
Steps per trajectory; final time
104 ; 400
Appendix
Table 1: Model and reference-trajectory settings. The damping value refers to the integrator setting; no numerical value for γ in Eq. ( 10 ) is inferred from it.
Figure 6: Microscopic lattice pattern and emergent Ising-like CDW field in a representative 40×40 configuration. Left: displacement Qi , whose alternating pattern tracks the local charge modulation through electron–lattice coupling. Right: staggered neighbor-density contrast ϕi=(−1)xi+yi(ni−41∑j∈Ninj) 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(q,t) of the staggered CDW field on a 40×40 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 q=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→128→96→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
ϵ=10−5 ; α 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(τ) predicted by five candidate models ( C1 – C5 ) 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 ( C5 ) and the model with spatially coupled recurrent gates ( C3 ) exhibit the closest agreement with the reference dynamics.
Model
GRU
hi in gates
hi in candidate
RMSE
C1
No
–
–
0.032842
C2
Yes
No
No
0.018433
C3
Yes
Yes
No
0.006224
C4
Yes
No
Yes
0.017147
C5
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→128→96→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.
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.
Yunhao Fan, Gia-Wei Chern
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
Learning in recurrent neural networks can fundamentally reshape their underlying dynamics, transforming initially chaotic activity into stable task-dependent behavior. We develop a non-equilibrium dynamical mean-field theory(DMFT) to describe this transition during learning. We show that a slow feedback-driven learning process generates an evolving effective feedback strength that drives the network through a transition from chaotic to stable dynamics defined by a bifurcation of the DMFT solution. By deriving the two-time correlation function throughout learning, we identify a critical feedback strength and a corresponding learning rate dependent critical time separating these regimes. The transition arises from the progressive deformation of an effective dynamical landscape by the growing learned feedback structure. Starting from the untrained state, the theory predicts the time evolution of the network output during training and shows quantitative agreement with numerical simulations.
Varun Vaidya
Department of Physics, University of South Dakota, Vermillion 57069, USA
Metallic magnets exhibit complex spin dynamics governed by electronically generated interactions. Predictive simulations of such dynamics typically require repeated solutions of an underlying electronic problem throughout the time evolution, creating a major computational bottleneck. Here we introduce a graph neural network (GNN) magnetic force-field framework that learns the effective magnetic energy functional governing itinerant spin dynamics directly from electronic calculations. Conceptually analogous to machine-learned interatomic potentials, the proposed framework enables efficient evaluation of spin torques while capturing the nonlinear and spatially extended interactions generated by itinerant electrons. We benchmark the method on representative metallic magnetic systems exhibiting collinear, noncollinear, and noncoplanar magnetic order. The learned force fields accurately reproduce electronically generated spin torques and yield nonequilibrium spin dynamics in excellent agreement with direct electronic simulations. Our results establish graph neural networks as a powerful framework for machine-learned magnetic force fields, providing a pathway toward predictive large-scale simulations of nonequilibrium magnetism across multiple length and time scales.
Ali Rayat, Yunhao Fan, Gia-Wei Chern
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA