A unified deep learning framework for nonlinear time-series forecasting: a case study of gold price data

Accurately forecasting gold prices remains a challenging task due to their highly nonlinear, non-stationary, and volatile dynamics, which involve complex multi-scale temporal dependencies and intricate inter-feature interactions. To address the limitations of existing deep learning (DL) approaches in effectively modeling such nonlinear characteristics, this study proposes a hybrid architecture, namely Convolutional Neural Network–Squeeze-and-Excitation–Bidirectional Gated Recurrent Unit (CNN–SE–BiGRU). The proposed model integrates convolutional feature extraction, nonlinear channel-wise attention recalibration, and bidirectional recurrent learning to enhance the representation of complex nonlinear temporal patterns. Using a real-world Indian gold price dataset spanning more than a decade and covering diverse market conditions, the proposed CNN–SE–BiGRU is benchmarked against a comprehensive set of baseline, hybrid, attention-enhanced, and ablation models under a unified experimental framework. The results demonstrate that the proposed model achieves the best overall performance, obtaining the lowest Normalized Root Mean Squared Error (NRMSE) of 0.095595 and the highest correlation coefficient (R) of 0.995430, indicating superior predictive accuracy and stability. Visual analyses further confirm its strong capability in tracking nonlinear price dynamics and maintaining consistent error behavior, while a nonparametric Bootstrap test with 10,000 iterations verifies that the observed performance improvements are statistically significant. These findings highlight the effectiveness of jointly modeling nonlinear temporal dependencies, inter-channel feature interactions, and bidirectional dynamics within a unified framework, establishing the CNN–SE–BiGRU model as a robust and reliable solution for nonlinear financial time-series forecasting.