From short-sighted to far-sighted: A comparative study of recursive machine learning approaches for open quantum systems

Accurately modeling the dynamics of open quantum systems is critical for advancing quantum technologies, yet traditional methods often struggle with balancing accuracy and efficiency. Machine learning (ML) offers a promising alternative, particularly through recursive models that predict system evolution based on the past history. While these models have shown success in predicting single observables, their effectiveness in more complex tasks, such as forecasting the full reduced density matrix (RDM), remains unclear. In this work, we extend history-based recursive ML approaches to complex quantum systems, comparing four physics-informed neural network (PINN) architectures: (i) single-RDM-predicting PINN (SR-PINN), (ii) SR-PINN with simulation parameters (PSR-PINN), (iii) multi-RDMs-predicting PINN (MR-PINN), and (iv) MR-PINN with simulation parameters (PMR-PINN). We apply these models to two representative open quantum systems: the spin-boson (SB) model and the Fenna-Matthews-Olson (FMO) complex. Our results demonstrate that single-RDM-predicting models (SR-PINN and PSR-PINN) are limited by a narrow history window, failing to capture the full complexity of quantum evolution and resulting in unstable long-term predictions, especially in nonlinear and highly correlated dynamics. In contrast, multi-RDMs-predicting models (MR-PINN and PMR-PINN) provide more accurate predictions by extending the forecast horizon, incorporating long-range temporal correlations, and mitigating error propagation. Surprisingly, including simulation parameters explicitly, such as temperature and reorganization energy, in PSR-PINN and PMR-PINN does not consistently improve accuracy and, in some cases, even reduces performance. This suggests that these parameters are already implicitly encoded in the RDM evolution, making their inclusion redundant and adding unnecessary complexity. These findings highlight the limitations of short-sighted recursive forecasting in complex quantum systems and demonstrate the superior stability and accuracy of far-sighted approaches for long-term predictions.