## 5 Conclusions  
This study introduces IGWO-MALSTM, a hybrid deep learning framework that combines Improved Grey Wolf Optimization (IGWO) and a multi-head attention-enhanced LSTM for financial time series forecasting. The model addresses two critical challenges in conventional LSTM-based forecasting: (1) the inefficiency and subjectivity of manual hyperparameter tuning, and (2) the limited ability to capture long-term temporal dependencies due to gradient-related issues. By enhancing the population initialization process using the Mersenne Twister in IGWO, the proposed approach significantly improves the optimizer's search diversity and convergence stability. Simultaneously, the integration of a multi-head attention mechanism allows the model to dynamically focus on salient temporal features across different \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Indicator** & **Rbreaker** & **Rbreaker-LA** &
\begin{tabular}{c} **Rbreaker-IGWO-MALSTM** \\ \end{tabular} \\ \hline Initial Capital & 50,000 & 50,000 & 50,000 \\ Ending Capital & 61,772.3 & 71,965.3 & 168,233.7 \\ Total Profit/Loss & 11,772.3 & 21,965.3 & 118,233.7 \\ Average Profit/Loss & 60.4 & 264.6 & 185.6 \\ Return Rate & 23.3\% & 43.9 & 212.3\% \\ Annualized Return Rate & 4.61\% & 8.78\% & 42.6\% \\ Total Number of Trades & 195 & 83 & 637 \\ Total Number of Profitable Trades & 118 & 50 & 387 \\ Average Profit & 214.4 & 839.5 & 1118.8 \\ Maximum Profit & 1217.1 & 3955.1 & 12,121.1 \\ Number of Losing Trades & 77 & 33 & 250 \\ Average Loss & \(-\)175.6 & \(-\)606.4 & \(-\)1259.0 \\ Maximum Loss & \(-\)1507.4 & \(-\)2595.9 & \(-\)7537.1 \\ Maximum Drawdown Ratio & 0.39\% & 1.38\% & 3.66\% \\ Maximum Drawdown Amount & 197.46 & 669.98 & 2209.66 \\ Sharpe Ratio & \(-\)1.9853 & \(-\)0.34 & 1.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of backtesting results. subspaces, thereby mitigating gradient vanishing/explosion and boosting long-term forecasting accuracy. Extensive experiments on both low-frequency (futures spread) and high-frequency (CSI 300 index) datasets demonstrate that IGWO-MALSTM consistently outperforms a wide range of benchmark models, including RNN, GRU, LSTM, PSO-LSTM, CS-LSTM, Transformer, and LSTM-Transformer. The model achieved up to 61.45% reduction in MSE and 44.53% reduction in MAE, with statistically significant improvements confirmed by \(t\)-tests and Wilcoxon tests. Moreover, the ablation study verified that both IGWO and multi-head attention contribute substantially to performance gains, and their combined use achieves optimal results. From a practical perspective, the model's effectiveness was further validated through backtesting on real futures trading data, where IGWO-MALSTM-integrated strategies significantly outperformed traditional and LSTM-enhanced trading algorithms in terms of return rate, Sharpe ratio, and risk-adjusted profitability. These findings not only demonstrate the model's predictive strength but also underscore its potential for real-world financial applications, particularly in algorithmic trading and quantitative investment strategies. Future work will focus on enhancing model generalization across asset classes, incorporating external macroeconomic indicators and sentiment analysis, and extending IGWO to support online learning or dynamic strategy adaptation in rapidly changing markets. M.Z., conceptualization, methodology, and writing--original draft and supervision; H.Q., writing--original draft and supervision; P.Q., formal analysis. All authors have read and agreed to the published version of the manuscript. This research was funded by Key Research Project of Henan Province (231111210500). \"Double first-class\" discipline creation Project of Surveying and Mapping Science and Technology: GCCYJKT202513. Not applicable. No new data were created or analyzed in this study. Data sharing is not applicable to this article. Authors [PERSON] and [PERSON] were employed by the company Hebi National Optoelectronic Technology Co., Ltd.