SMD-xLSTM-TSMixer: A deep learning framework with wave-age modulation for wave height forecasting

Accurate wave height prediction is crucial for maritime safety and coastal protection, while reliable long-term forecasts are essential for infrastructure design and energy management. Existing models primarily rely on historical wave height data as the sole input, neglecting the physical dynamics of wave generation. In this paper, a novel SMD-xLSTM-TSMixer model is proposed. Physical domain knowledge is integrated by incorporating wave age and wave steepness as auxiliary variables. To mitigate data non-stationarity, a hybrid SMD decomposition strategy is constructed, combining Seasonal-Trend Decomposition using LOESS (STL) with Variational Mode Decomposition (VMD). The decomposed sub-signals are subsequently processed by a dual-branch neural network. Specifically, long-term temporal dependencies are captured by an xLSTM, while complex inter-channel correlations are extracted by a TSMixer. Finally, features from these parallel branches are adaptively integrated via a gating mechanism. Experimental results on four buoy datasets show that, compared to effective benchmark models such as VMD-LSTM-TCN and SMD-LSTM-Informer, the proposed model reduces error metrics (RMSE, MAE, and MAPE) by more than 50% within a 24-h prediction horizon. Furthermore, in a 48-h prediction scenario, the coefficient of determination exceeds 0.92 for all datasets. This framework demonstrates stability and robustness in extreme wave events, overcoming peak-shaving and time-lag issues. • Integrates physical knowledge of wave age and steepness to overcome limitations of single inputs. • A hybrid decomposition strategy combining STL and VMD effectively extracts multi-scale features. • Parallel architecture integrating xLSTM and TSMixer captures temporal and inter-channel dependencies. • Overcomes peak-shaving and time-lag issues, reducing errors by >50% against benchmarks. • Achieves robust long-term performance with R 2>0.92 at a 48-h prediction scenario.