## 5 Experimental Results
### Qualitative Results
Adjusting the background error variances using the DRL-AST's adaptive variance rescaling strategy as an initial state enables effective model prediction for nearly 7 days. In comparison, other strategies can barely maintain about 2 days of effective model prediction before deviating markedly from true values and being considered invalid predictions. This result demonstrates that DRL-AST notably extends the effective forecast lead time by adaptively optimizing **B**. Unlike other variance rescaling methods, DRL-AST leverages a GRU module to analyze historical assimilation states, dynamically adjusting **B** to enhance consistency between the analysis 3D-Var and the model's dynamical constraints. This adaptive approach helps delay error amplification, allowing the model to maintain skillful predictions for nearly 7 days. However, the inherent instability of chaotic systems, characterized by exponential error growth within the Lyapunov timescale, imposes a fundamental limit on forecast reliability. While DRL-AST refines initial conditions to enhance short-term  
Figure 7: Deviation between the forecast state \(\mathbf{x}^{\prime}\) and the true state \(\mathbf{x}^{\prime}\) for \(\sigma_{\psi^{\prime}}=1.0\). Panels **(a–d)** correspond to NO, CON, CUTE-ST, and DRL-AST, respectively. The y-axis represents the indices of the 40 variables in the Lorenz-96 model, while the x-axis denotes the forecast lead time in Lyapunov units (up to 8 units). The green box highlights the Lyapunov time range within which the forecasts remain accurate. predictability, its impact on decadal or longer-term predictions remains uncertain, as initial condition effects diminish beyond a decade, and model biases dominate [67]. Additionally, incorporating geometric attractor theory and chaos-based methodologies offers valuable perspectives for understanding DRL-AST's role in long-term predictions. Attractors define the long-term evolution of chaotic systems in phase space, and optimizing initial conditions through DRL-AST may enable numerical models to better align with the intrinsic attractor dynamics [68], thereby improving the stability of long-term statistical predictions. Furthermore, future research could explore the application of DRL-AST in seasonal-to-decadal (1-10 years) prediction, integrating classical and quantum neural network (QNN) modeling approaches to enhance forecast stability and generalization. QNNs exhibit promise in high-dimensional climate system modeling by refining initial conditions and mitigating error growth, thereby strengthening long-term forecast reliability while facilitating effective integration with physical models [69]. Figure 8: Single-state variable (\(x10,x20,x30\), and \(x40\)) from assimilation analysis states to model forecast. \(\sigma_{y^{\prime}}=0.5\) for Figures **(a–d)** and \(\sigma_{y^{\prime}}=1.5\) for Figures **(e–h)**. The black thick solid lines are the model’s true values; red thick solid lines denote DRL-AST analysis/forecast; thin orange solid lines denote NO analysis/orange; thin yellow-green solid lines denote CON analysis/forecast; thin purple solid lines denote CUTE-ST analysis/forecast; sky-blue diamond dots denote observations. Thirty-day assimilation to the left of the dotted line, and fifteen-day model forecast to the right. The inflection points are highlighted in red boxes. Figure 9: Magnified view of forecast trajectories around the inflection points marked in Figure 8a–d. Subfigures **(i)**–**(iv)** correspond to the red-highlighted regions in Figure 8a–d, respectively, and provide refined comparisons of the forecast behavior. Line styles and color schemes follow those used in Figure 8.