加权布爾霍夫平均加速數據驅動方法收斂，提升準確性且無額外開銷。
Weighted Birkhoff Averages Accelerate Data-Driven Methods
- 對布爾霍夫平均進行末端加權，改變傳統算法收斂慢的問題。
- 在流體與厄爾尼諾數據上，收斂速度達超多項式甚至指數級提升。
- 可直接嵌入現有方法，適用於動力系統分析、數據預測等場景。
動力系統中的許多數據驅動算法依賴於收斂極慢的遍歷平均。一個簡單思路改變了這一狀況：對端點進行加權。加權布爾霍夫平均可實現更快收斂（有時為超多項式甚至指數級），並能無縫融入現有方法。本文展示了五種加權算法：加權動態模態分解（wtDMD）、加權擴展DMD（wtEDMD）、加權稀疏非線性動力學識別（wtSINDy）、加權譜測度估計與加權擴散預測。在從流體流動到厄爾尼諾數據的多個案例中，結果清晰表明：加權幾乎不增加成本，實現簡單，卻能顯著提升相同數據下的性能。
原文摘要 · Abstract (English)
Many data-driven algorithms in dynamical systems rely on ergodic averages that converge painfully slowly. One simple idea changes this: taper the ends. Weighted Birkhoff averages can converge much faster (sometimes superpolynomially, even exponentially) and can be incorporated seamlessly into existing methods. We demonstrate this with five weighted algorithms: weighted Dynamic Mode Decomposition (wtDMD), weighted Extended DMD (wtEDMD), weighted Sparse Identification of Nonlinear Dynamics (wtSINDy), weighted spectral measure estimation, and weighted diffusion forecasting. Across examples ranging from fluid flows to El Niño data, the message is clear: weighting costs nothing, is easy to implement, and often delivers markedly better results from the same data.
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