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Communications in Computational Chemistry

Two-Dimensional Convolutional Neural Networks for Predicting Non-Markovian Quantum Dissipative Dynamics

Zhiqian Huang, Cun Long, Qingdong Zhu, Jiaan Cao, Xiao Zheng

Published Jun 1, 2025• 0 citations
Publisher

Abstract

Simulating non-Markovian quantum dissipative dynamics remains a major challenge in theoretical and computational chemistry. While traditional numerical methods such as hierarchical equations of motion are numerically exact, they suffer from prohibitive computational costs when modeling systems with complex environmental couplings or strong non-Markovian effects. To address this limitation, we propose a deep learning framework based on two-dimensional convolutional neural networks (2D-CNN) for efficiently predicting long-time quantum dissipative dynamics using only short-time trajectory data. Our approach incorporates a 1D-to-2D feature reconstruction strategy, which transforms 1D time-series data into 2D images, and a multi-timescale fusion network to resolve complex dynamical features. We validate the framework on two paradigmatic cases — dissipative relaxation in a two-level system and Rabi oscillations in a dissipative spin system — achieving prediction mean absolute errors of $10^{-3}$ and $10^{-2},$ respectively. The results highlight the effectiveness of our 2D-CNN approach in capturing long-time temporal correlations, providing a computationally efficient pathway for simulating quantum dynamics in realistic open systems.

DOI
10.4208/cicc.2025.90.01

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Simulating non-Markovian quantum dissipative dynamics remains a major challenge in theoretical and computational chemistry. While traditional numerical methods such as hierarchical equations of motion are numerically exact, they suffer from prohibitive computational costs when modeling systems with complex environmental couplings or strong non-Markovian effects. To address this limitation, we propose a deep learning framework based on two-dimensional convolutional neural networks (2D-CNN) for efficiently predicting long-time quantum dissipative dynamics using only short-time trajectory data. Our approach incorporates a 1D-to-2D feature reconstruction strategy, which transforms 1D time-series data into 2D images, and a multi-timescale fusion network to resolve complex dynamical features. We validate the framework on two paradigmatic cases — dissipative relaxation in a two-level system and Rabi oscillations in a dissipative spin system — achieving prediction mean absolute errors of 10−310^{-3}10−3 and 10−2,10^{-2},10−2, respectively. The results highlight the effectiveness of our 2D-CNN approach in capturing long-time temporal correlations, providing a computationally efficient pathway for simulating quantum dynamics in realistic open systems.
The non-Markovian stochastic Schrödinger equation (NMSSE) offers a promising approach for open quantum simulations owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end, we propose a scheme that utilizes neural networks to extract effective information from long-time simulations of the non-Markovian stochastic Schrödinger equation, fine-tunes the networks using converged short-time data, and predicts long-term system evolution, eliminating the influence of stochastic oscillations. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs), along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thus substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.
Exact numerical simulations of dynamics of open quantum systems often require immense computational resources. We demonstrate that a deep artificial neural network composed of convolutional layers is a powerful tool for predicting long-time dynamics of open quantum systems provided the preceding short-time evolution of a system is known. The neural network model developed in this work simulates long-time dynamics efficiently and accurately across different dynamical regimes from weakly damped coherent motion to incoherent relaxation. The model was trained on a data set relevant to photosynthetic excitation energy transfer and can be deployed to study long-lasting quantum coherence phenomena observed in light-harvesting complexes. Furthermore, our model performs well for the initial conditions different than those used in the training. Our approach reduces the required computational resources for long-time simulations and holds the promise for becoming a valuable tool in the study of open quantum systems.
The Non-Markovian Stochastic Schrodinger Equation (NMSSE) offers a promising approach for open quantum simulations, especially in large systems, owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end,we propose a scheme that combines neural network techniques with simulations of the non-Markovian stochastic Schrodinger equation. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs),along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thereby substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.
In this Communication, we demonstrate that a deep artificial neural network based on a transformer architecture with self-attention layers can predict the long-time population dynamics of a quantum system coupled to a dissipative environment provided that the short-time population dynamics of the system is known. The transformer neural network model developed in this work predicts the long-time dynamics of spin-boson model efficiently and very accurately across different regimes, from weak system-bath coupling to strong coupling non-Markovian regimes. Our model is more accurate than classical forecasting models, such as recurrent neural networks, and is comparable to the state-of-the-art models for simulating the dynamics of quantum dissipative systems based on kernel ridge regression.
Recurrent neural networks (RNNs) have recently been extensively applied to model the time evolution in fluid dynamics, weather predictions, and even chaotic systems due to their ability to capture temporal dependencies and sequential patterns in data. Here we present an RNN model based on convolutional neural networks for modeling the nonlinear nonadiabatic 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 semiclassical 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.
Reducing computational scaling for simulating non-Markovian dissipative dynamics using artificial neural networks is both a major focus and formidable challenge in open quantum systems. To enable neural quantum states (NQSs), we encode environmental memory in dissipatons (quasiparticles with characteristic lifetimes), yielding the dissipaton-embedded quantum master equation (DQME). The resulting NQS-DQME framework achieves compact representation of many-body correlations and non-Markovian memory. Benchmarking against numerically exact hierarchical equations of motion confirms NQS-DQME maintains comparable accuracy while enhancing scalability and interpretability. This methodology opens new paths to explore non-Markovian open quantum dynamics in previously intractable systems.