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.
Figures & tables
Figure 1: Schematic of the BP-type force-field architecture for the semiclassical Holstein model. Starting from the input lattice distortion configuration {Qi} (left), the local environment within a cutoff radius rc around a target site i is first processed through a symmetry-adapted descriptor. The descriptor maps the local configuration to a set of generalized coordinates {Gm} , constructed such that all eight symmetry-related configurations generated by the D4 point-group operations of the square lattice yield identical descriptor values. These symmetry-invariant descriptors {Gm} are then fed into a fully connected neural network, which outputs a single scalar Fi corresponding to the force acting on site i .
Layer
Network
Input (descriptor) layer
FC (45,4096) 1 1 1 Fully Connected (FC) layer with arguments (input size, output size), corresponding to Eq. ( 11 ). σ 2 2 2 Activation function used in FC layer. = ReLU
Hidden Layer 1
FC (4096,2048)σ = ReLU
Hidden Layer 2
FC (2048,750)σ = ReLU
Hidden Layer 3
FC (750,592)σ = ReLU
Hidden Layer 4
FC (592,256)σ = ReLU
Hidden Layer 5
FC (256,127)σ = ReLU
Table 1: Architecture and hyperparameter details of the multilayer perceptron (MLP) model.
Figure 2: Force prediction benchmark for the descriptor+MLP model. (a) Parity plot comparing the machine-learning predicted force FML with the exact diagonalization result FED for both training (blue) and test (orange) datasets. The near-perfect alignment along the diagonal demonstrates excellent predictive accuracy and generalization. (b) Distribution of the force prediction error δ=FML−FED . The error histogram is sharply peaked around zero, indicating unbiased predictions with small variance and high numerical fidelity.
Figure 3: Schematic architecture of the graph neural network (GNN) for learning the adiabatic dynamics of the Holstein model. Each layer of the network corresponds to a lattice representation of the physical system. The input layer consists of the on-site lattice distortion configuration Qi defined on lattice sites. At each hidden layer, the lattice sites carry multi-channel scalar node features Vi,α(ℓ) that encode progressively more nonlocal information through successive message-passing operations. The output layer produces the predicted forces Fi acting on each lattice site. The message-passing update at each layer is illustrated schematically as a single-layer fully connected neural network acting on the local neighborhood of a site, combining same-site contributions ( W(self) ) and neighbor contributions ( W(NN) ) with shared parameters across the lattice. This locality-preserving and weight-sharing structure ensures translation-equivariant predictions and scalability to large lattice sizes.
Layer
Network
Input (embedding) layer
GCN(1,128) 3 3 3 A Graph Convolutional Network (GCN) layer with arguments (input size, output size) implements the message-passing and node-update operations defined in Eqs. ( 15 ) and ( 16 ). σ 4 4 4 The activation function used in GCN. = ReLU
Hidden Layer 1
GCN(128,512) σ = ReLU
Hidden Layer 2
GCN(512,1024) σ = ReLU
Hidden Layer 3
GCN(1024,2048) σ = ReLU
Hidden Layer 4
GCN(2048,1024) σ = ReLU
Hidden Layer 5
GCN(1024,512) σ = ReLU
Table 2: Architecture and hyperparameter details of the graph neural network (GNN) for direct force prediction of the Holstein model.
Figure 4: Force prediction benchmark for the GNN model. (a) Parity plot comparing the machine-learning predicted forces FML with the exact-diagonalization reference values FED for both training (blue) and test (orange) datasets. The close alignment of data points along the diagonal indicates high predictive accuracy and good generalization. (b) Distribution of the force prediction error δ=FML−FED . The error histogram is narrowly centered around zero, demonstrating unbiased predictions with small variance and high numerical accuracy.
Layer
Network
Input layer
Node feature dimension = 1
Message-passing block 1
Aggregate MLP (1,512) 5 5 5 Aggregate MLP with arguments (input channels, output channels) implements Eq. ( 19 ). Update MLP (1+512→512) 6 6 6 Update MLP with arguments (input channels, output channels) implements Eq. ( 20 ). σ 7 7 7 The activation function used in MLP(⋅) and Eq. ( 20 ). = ReLU Residual projection (1→512) 8 8 8 The residual projection implements Eq. ( 21 ).
Table 3: Message-passing neural network (MPNN) architecture and hyperparameters used in this work.
Figure 5: Force prediction benchmark for the energy-based GNN model. (a) Parity plot of machine-learning forces FML —obtained from automatic differentiation of the predicted energy—versus exact-diagonalization values FED for training (blue) and test (orange) data. The close alignment along the diagonal indicates high accuracy and good generalization. (b) Distribution of the force error δ=FML−FED , sharply centered at zero, demonstrating unbiased and low-variance predictions.
Figure 6: Dynamical benchmark and large-scale quench simulations using the direct-force GNN model. (a) Comparison of the ensemble-averaged equal-time correlation function ⟨QiQj⟩ at three representative times after a thermal quench, obtained from exact diagonalization (ED)–based Langevin dynamics and machine-learning (ML)–based Langevin dynamics on a 40×40 lattice. (b) Large-scale thermal quench simulations on a 200×200 lattice using the trained GNN force field, showing the real-space evolution and coarsening of charge-density-wave (CDW) domains at increasing time steps. The domain size grows systematically with time, illustrating curvature-driven coarsening dynamics. (c) Corresponding ensemble-averaged structure factor S(q,t) at the same times as in (b).
Figure 7: (a) Ensemble-averaged equal-time correlation function C(r,t)=⟨ϕiϕj⟩ of the local CDW order parameter ϕi , plotted versus the rescaled distance r/L(t) at several times after the quench. The data collapse onto a single master curve demonstrates dynamical scaling governed by a single characteristic length scale L(t) . (b) Time evolution of the domain size (correlation length) L(t) for three temperatures. The growth follows an approximate power law, L(t)∼tα , with a temperature-dependent exponent α consistently smaller than the Allen-Cahn value 1/2 .
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
Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both Z2 and U(1) lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in U(1) quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.
Ali Rayat, Gia-Wei Chern
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
We introduce implicit machine learning force fields (I-MLFFs), which replace explicit stacks of neural network layers with self-consistent fixed-point equations. In molecular simulations, this formulation enables intermediate representations to be reused across successive timesteps, thereby warm-starting force evaluation. The resulting models effectively combine the computational footprint of a shallow, single-layer MLFF with the representational capacity and accuracy of a deep neural network. Our approach unlocks architecture-agnostic efficiency gains that are inaccessible when force prediction and trajectory integration are considered separately. We demonstrate this across three major classes of graph neural networks: invariant, equivariant Cartesian tensor, and SO(3)-equivariant spherical-tensor architectures. Each yields a two- to five-fold reduction in compute and memory footprint. Crucially, these gains are achieved while retaining full atomistic resolution and the original integration timestep, avoiding spatial or temporal coarse graining. Our contribution therefore advances the scaling frontier of quantum-mechanically faithful molecular simulation, enabling longer trajectories and larger atomistic systems within fixed GPU memory and compute budgets, and thereby opening access to new insights across biomolecular and material systems.
Johannes Maeß, Leon Werner, J. Thorben Frank +5
BIFOLD, Berlin, Germany. · Machine Learning Group, TU Berlin, Berlin, Germany. · Department of Artificial Intelligence, Korea University, Seoul, South Korea. +1