Seasonal–trend-decomposed transformer has empowered long-term time series forecasting via capturing global temporal dependencies (e.g., period-based dependencies) in disentangled temporal patterns. However, existing methods design various auto-correlation or attention mechanisms in the seasonal view while ignoring the fine-grained temporal patterns in the trend view in the series decomposition component, which causes an information utilization bottleneck. To this end, a Transformer-based seasonal–trend decomposition methodology with a multi-scale attention mechanism in the trend view and a multi-view attention mechanism in the seasonal view is proposed, called DESTformer. Specifically, rather than utilizing the moving average operation in obtaining trend data, a frequency domain transform is first applied to extract seasonal (high-frequency) and trend (low-frequency) components, explicitly capturing different temporal patterns in both seasonal and trend views. For the trend component, a multi-scale attention mechanism is designed to capture fine-grained sub-trends under different receptive fields. For the seasonal component, instead of the frequency-only attention mechanism, a multi-view frequency domain (i.e., frequency, amplitude, and phase) attention mechanism is designed to enhance the ability to capture the complex periodic changes. Extensive experiments are conducted on six benchmark datasets covering five practical applications: energy, transportation, economics, weather, and disease. Compared to the state-of-the-art FEDformer, our model shows reduced MSE and MAE by averages of 6.5% and 3.7%, respectively. Such experimental results verify the effectiveness of our method and point out a new way towards handling trends and seasonal patterns in long-term time series forecasting tasks.

The dwell time algorithm is one of the key technologies that determines the accuracy of a workpiece in the field of ultra-precision computer-controlled optical surfacing. Existing algorithms mainly consider meticulous mathematics theory and high convergence rates, making the computation process more uneven, and the flatness cannot be further improved. In this paper, a reasonable elementary approximation algorithm of dwell time is proposed on the basis of the theoretical requirement of a removal function in the subaperture polishing and single-peak rotational symmetry character of its practical distribution. Then, the algorithm is well discussed with theoretical analysis and numerical simulation in cases of one-dimension and two-dimensions. In contrast to conventional dwell time algorithms, this proposed algorithm transforms superposition and coupling features of the deconvolution problem into an elementary approximation issue of function value. Compared with the conventional methods, it has obvious advantages for improving calculation efficiency and flatness, and is of great significance for the efficient computation of large-aperture optical polishing. The flatness of φ150 mm and φ100 mm workpieces have achieved PVr₁₅₀ = 0.028 λ and PVcr₁₀₀ = 0.014 λ respectively.