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Graph-Recurrent Learning of Emergent Collective Dynamics
Ali Rayat, Acacia Lee, Gia-Wei Chern
TL;DR
After unresolved microscopic degrees of freedom are eliminated, the observable field need not have closed, memoryless dynamics, complicating the description of evolving collective order. The GRNN learns finite-time evolution directly from trajectories by jointly updating observables and recurrent hidden states coupled through graph messages. For CDW ordering, the GRNN reproduces KPM temporal correlations, spatial correlation profiles and scaling collapse, and algebraic domain growth L(t) ∼ t^α with α < 1/2.
Problem
After unresolved microscopic degrees of freedom are eliminated, the observable field need not have closed, memoryless dynamics, complicating the description of evolving collective order.
Source: pp. 1, 4Method
The GRNN learns finite-time evolution directly from trajectories by jointly updating observables and recurrent hidden states coupled through graph messages.
Source: pp. 1–2Results
For CDW ordering, the GRNN reproduces KPM temporal correlations, spatial correlation profiles and scaling collapse, and algebraic domain growth L(t) ∼ t^α with α < 1/2.
Source: p. 4Takeaways & Limitations
The framework provides a learned reduced description of collective ordering that retains temporal memory and spatial coupling without specifying microscopic equations of motion.
Source: pp. 1, 5The demonstration is adiabatic; nonadiabatic or time-dependent Hartree–Fock applications may require larger hidden states or more expressive memory to encode unresolved electronic dynamics.
Source: pp. 3, 5
Abstract
from arXiv · showhide
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.
Appendix A: Graph-recurrent update
The GRNN advances the physical field and site-wise hidden states together through a shared update. Graph layers and recurrent messages let spatial features and neighboring memory influence each site's prediction.
Appendix A: Graph-recurrent update
- Each site’s observable and hidden state evolve together through a shared learned update, with hidden states retaining information from earlier configurations.The next sampled configuration is separated by the physical prediction interval Δt_pred.Source: Appendix A · p. 5
- Graph layers build spatial features from the current observable field, while the recurrent update also uses local and neighboring hidden states.The reset gate acts on local previous memory; neighboring memory enters through a separate term.Source: Appendix A · p. 5, Figure 5 · p. 6
- The readout uses the spatial embedding and updated memory to predict a change that is applied to the current physical field.Repeated recurrent steps let hidden information move farther across the lattice than a single graph-layer neighborhood.Source: Appendix A · p. 6
Appendix B: Holstein reference dynamics and CDW observables
The appendix describes the adiabatic Holstein reference dynamics and the staggered CDW field used to represent its order. It also defines a momentum-space view of coarsening that the GRNN reproduces against KPM trajectories.
Appendix B: Holstein reference dynamics and CDW observables
- At fixed lattice displacement, the electronic density determines the force on classical lattice coordinates; KPM evaluates the electronic contribution as the lattice evolves.The classical dynamics are integrated with a Verlet-type scheme.Source: Appendix B · p. 6
- The staggered density contrast removes checkerboard alternation, yielding an Ising-like field whose sign distinguishes the two ordered CDW domains.The displacement pattern tracks local charge modulation through electron–lattice coupling.Source: Appendix B · p. 6, Figure 6 · p. 7
- The equal-time structure factor gives a momentum-space view of coarsening, with its staggered-field ordering peak centered at q = 0.The corresponding peak in the density structure factor is at (π, π).Source: Appendix B · pp. 6–7
- As spatial correlations extend to larger distances, the broad initial structure-factor peak narrows, and GRNN reproduces the evolution of the KPM momentum-space shape shown in maps at three times.Source: Appendix B · p. 7
Appendix C: Kernel Polynomial Method for Reference Data Generation
The appendix explains how the reference trajectories are generated efficiently with the kernel polynomial method. Chebyshev expansions approximate functions of the rescaled electronic Hamiltonian, and the resulting forces drive lattice dynamics.
Appendix C: Kernel Polynomial Method for Reference Data Generation
- KPM approximates functions of the single-particle Hamiltonian with a Chebyshev expansion, avoiding full diagonalization when generating reference trajectories.The electronic forces computed this way are incorporated into the Langevin equations for the classical lattice.Source: Appendix C · pp. 7–8
- The Hamiltonian is rescaled so its spectrum lies in [−1, 1] before functions of it are approximated by a truncated Chebyshev series.The expansion order and coefficients, including damping factors, specify the approximation.Source: Appendix C · pp. 7–8
- Chebyshev polynomials are evaluated recursively, requiring only sparse matrix-vector multiplications.At fixed expansion order and sampling parameters, this gives cost linear in the number of lattice sites.Source: Appendix C · p. 8
- The resulting microscopic trajectories are mapped to the local CDW order parameter for GRNN training and reference benchmarks.Figure 7 compares KPM reference trajectories with autoregressive GRNN predictions; the predicted momentum-space peak narrows as domains grow.Source: Appendix C · p. 8, Figure 7 · p. 8
Appendix D: Network implementation and training for the Holstein demonstration
The Holstein implementation uses shared graph, recurrent, and readout parameters at every lattice site, with a bounded readout for the scalar CDW field. Training uses trajectory sequences, burn-in, and truncated backpropagation through time.
Appendix D: Network implementation and training for the Holstein demonstration
- The model uses three graph aggregation layers and a sitewise MLP readout, with the same parameters at every site so parameter count does not grow with lattice area.The graph dimensions are 1 → 128 → 96 → 64; the readout has four hidden layers.Source: Appendix D · pp. 8–9
- Because the scalar CDW field is bounded by −1 ≤ ϕ_i ≤ 1, its readout applies the learned change in transformed coordinates while keeping the output within range.The readout uses ϵ = 10^-5 and a learned scale α initialized to 0.1; this specialization is not required by the general GRNN update.Source: Appendix D · p. 9
- The architecture and training settings for the Holstein demonstration are listed in Table II.Complete trajectories, rather than individual snapshots, are assigned to training, validation, and test sets.Source: Table II · p. 9, Appendix D · p. 9
- Hidden states start at zero and are conditioned on reference configurations during burn-in, which is excluded from the trajectory mean-squared-error loss.After burn-in, reference fields are supplied with probability p = 0.2 during training; validation and testing proceed autoregressively after burn-in.Source: Appendix D · p. 9
- Truncated backpropagation carries predicted fields and hidden states across chunks but detaches them at chunk boundaries, limiting gradient propagation to each chunk.Gradients are accumulated across chunks before parameter updates.Source: Appendix D · p. 9
Appendix E: Ablation Study of Candidate Models
The ablation compares five GRNN variants to assess recurrent memory and spatially aggregated hidden states. The full GRNN has the lowest temporal-autocorrelation RMSE, while spatially coupled recurrent gates account for most of the improvement.
Appendix E: Ablation Study of Candidate Models
- All candidates share the same GNN backbone and training procedure; their autoregressive temporal autocorrelations are compared with the KPM reference.Training uses 50-step sequences, 20-step burn-in, and 200 epochs.Source: Appendix E · pp. 9–10
- 6.15 × 10−3 RMSE: full GRNN (C5) achieves the lowest error, with only a relatively small improvement over C3.Table III summarizes RMSE relative to the KPM reference.Source: Table III · p. 10, Appendix E · p. 11
- C1 has no recurrent connections; C2 uses local GNN embeddings and hidden states, while C3–C5 incorporate spatially aggregated hidden states into different recurrent components.C3 adds spatial information to gates, C4 to the candidate update, and C5 to both.Source: Appendix E · p. 10
- 1.84×10−2 RMSE for recurrent C2 improves on the memoryless GNN C1’s 3.28×10−2, supporting the value of local recurrent memory.Source: Appendix E · pp. 10–11
- 6.22×10−3 RMSE for C3 versus 1.71×10−2 for C4: spatially aggregated hidden states in recurrent gates outperform their use only in the candidate update.The authors identify spatial coupling in the update and reset gates as particularly important for reproducing temporal correlations.Source: Appendix E · p. 11