Weighted Birkhoff averages accelerate data-driven methods

doi:10.1098/rspa.2025.0979

Weighted Birkhoff averages accelerate data-driven methods

2026-03-01

0

PRE

AI

M

Maria Bou-Sakr-El-Tayar

B

Bramburger, Jason J. *

C

Colbrook, Matthew

DOI：10.1098/rspa.2025.0979

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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 Ni & ntilde;o data, the message is clear: weighting costs nothing, is easy to implement and often delivers markedly better results from the same data.

Keywords:

data-driven dynamical systems

Birkhoff averages

Koopman operators

DMD

model identification

spectral convergence

forecasting

Weighted Birkhoff averages accelerate data-driven methods

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