Recurrent convolutional neural networks for modeling non-adiabatic dynamics of quantum-classical systems

TL;DR

The paper addresses modeling non-adiabatic quantum–classical dynamics by combining physics-informed neural networks with recurrent convolutional architectures. It demonstrates that a standard CNN can accurately predict deterministic, shallow-quench dynamics in a 1D Holstein model, while a physics-aware PARC framework with differentiator–integrator CNNs can reproduce the statistical climate of chaotic deep-quench dynamics. The key contributions are (i) a CNN-based baseline for shallow quenches, (ii) a PARC-based approach that enforces a physics-like time-stepping structure to capture long-horizon statistics, and (iii) quantitative validation via trajectory tracking and autocorrelation analyses. The results highlight the potential of PARC to learn physically consistent dynamics in hybrid quantum–classical systems and motivate extensions to larger systems and stochastic surface-hopping dynamics for practical quantum chemistry and materials applications.

Abstract

Recurrent neural networks (RNNs) have recently been extensively applied to model the time-evolution in fluid dynamics, weather predictions, and even chaotic systems thanks to their ability to capture temporal dependencies and sequential patterns in data. Here we present a RNN model based on convolution neural networks for modeling the nonlinear non-adiabatic dynamics of hybrid quantum-classical systems. The dynamical evolution of the hybrid systems is governed by equations of motion for classical degrees of freedom and von Neumann equation for electrons. The physics-aware recurrent convolution (PARC) neural network structure incorporates a differentiator-integrator architecture that inductively models the spatiotemporal dynamics of generic physical systems. We apply our RNN approach to learn the space-time evolution of a one-dimensional semi-classical Holstein model after an interaction quench. For shallow quenches (small changes in electron-lattice coupling), the deterministic dynamics can be accurately captured using a single-CNN-based recurrent network. In contrast, deep quenches induce chaotic evolution, making long-term trajectory prediction significantly more challenging. Nonetheless, we demonstrate that the PARC-CNN architecture can effectively learn the statistical climate of the Holstein model under deep-quench conditions.

Figures (10)

• Figure 1: Schematic diagram of the recurrent structure for non-adiabatic dynamics of the semi-classical Holstein model. The framework is based on a single CNN for all three dynamical degrees of freedom: $Q$, $P$, and $\rho$.
• Figure 2: Architecture of the standard CNN model. The components $\rho$, $Q$, and $P$ are first prepared for input. $\rho$ is split into real-valued components $\rho_\text{real}$ and $\rho_\text{imag}$. $Q$ and $P$ are both inserted into the diagonals of zero-matrices. The resulting $4$-channel grid is then input into the CNN model with modified ResNet-v2 based architecture. $\rho$, $Q$, and $P$ are then extracted from the output of the CNN in the reverse manner they were input.
• Figure 3: Snapshots of phonon displacements $Q_i$, momentum $P_i$, and electron density matrix $\rho_{ij}$ at various times after a shallow quench from $g_i = 0.5$ to $g_f = 0.8$ of the semi-classical Holstein model on a $L = 16$ chain. The ground truth on the top row is obtained from the 4th-order Runge-Kutta integration of the governing dynamics Eqs. (\ref{['eq:newton_eq']}) and (\ref{['eq:von-neumann']}). The bottom row shows predictions from the trained simple CNN model. Only the real component of the complex-valued $\rho$ is shown. The steps are in units of prediction time steps, which for the shallow quench case is set to $\Delta t = 0.64$ time units. The model is given the system state at prediction step $0$ and predicts for the following $48$ prediction steps.
• Figure 4: Graphed ground truth (blue) vs predicted (red) $\Delta_\rho$ and $\Delta_Q$ for a shallow quench trajectory with system size $32$. The red prediction essentially perfectly covers the blue ground truth, which is under the red prediction in the visualization.
• Figure 5: Schematic diagram of the recurrent structure for non-adiabatic dynamics of the semi-classical Holstein model. The framework is based on a single PARC-based CNN differentiator (diff)/integrator (int) pair for all three dynamical degrees of freedom: $Q$, $P$, and $\rho$.