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.

翻译：动力系统中许多数据驱动算法依赖于遍历平均，其收敛速度极其缓慢。一个简单的想法改变了这一状况：对两端进行渐缩处理。加权Birkhoff平均能够实现更快的收敛速度（有时达到超多项式甚至指数级），并且可以无缝集成到现有方法中。我们通过五种加权算法证明了这一点：加权动态模态分解（wtDMD）、加权扩展动态模态分解（wtEDMD）、加权非线性动力学稀疏辨识（wtSINDy）、加权谱测度估计以及加权扩散预测。从流体流动到厄尔尼诺数据的一系列示例表明：加权处理无需额外成本，易于实现，且通常能在相同数据基础上获得显著改善的结果。