Weighted Birkhoff Averages Accelerate Data-Driven Methods
This addresses a key bottleneck in dynamical systems analysis for researchers, offering an easy-to-implement improvement that enhances existing methods without additional cost.
The paper tackles the slow convergence of ergodic averages in data-driven dynamical systems by introducing weighted Birkhoff averages, which can converge much faster (sometimes superpolynomially or exponentially) and are applied to five algorithms, showing markedly better results across examples like fluid flows and El Niño data.
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.