Weighted Birkhoff Averages Accelerate Data-Driven Methods

TL;DR

The paper tackles slow convergence of ergodic averages in data-driven dynamical analysis and introduces Weighted Birkhoff Averages (WBAs), tapering time sums to accelerate convergence while preserving the same limiting mean. By defining a smooth bump weight class $\mathcal{W}$ and the normalized ${WB_N(g)(x)}$, the authors show accelerated rates (often exponential or superpolynomial) across various dynamical regimes. They demonstrate five weighted methods—wtDMD, wtEDMD, wtSINDy, weighted spectral measures, and weighted diffusion forecasting—on canonical flows and the El-Niño dataset, consistently achieving higher accuracy from the same data and, in favorable regimes, substantial improvements. The work provides practical guidelines for integrating WBAs into operator learning, forecasting, and system identification, with open-source code and data to facilitate adoption.

Abstract

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.

Figures (11)

• Figure 1: Dynamics (top) and averages (bottom) of the driven logistic map \ref{['DrivenLogistic']}. Weighted (red) and unweighted (blue) time averages of the state variable $x$ are presented as a log-log plot of absolute errors against the average with $N = 10^6$. Parameter values are $\varepsilon = 0$ (left, periodic), $\varepsilon = 0.01$ (middle, quasiperiodic), and $\varepsilon = 0.1$ (right, chaotic).
• Figure 2: Left: DMD eigenvalues for $N = 1000$ snapshots of vorticity data from flow around a cylinder with $r = 11$ (top) and $r = 21$ (bottom). Middle and Right: Relative errors between the DMD (blue) and wtDMD (red) matrices (middle) and eigenvalues (right) over a range of snapshots $N = 10,20,\dots,500$ and the benchmark results using $N = 1000$ snapshots. The top panels use $r = 11$, while the bottom use $r = 21$.
• Figure 3: Averaged relative norm errors between the EDMD (blue) and wtEDMD (red) over a range of snapshots $N$ and the benchmark results using $N = 10^7$ snapshots. All results use data from the standard map \ref{['StandardMap']} with: (a) $\lambda = 0.5$; (b) $\lambda = 0.75$; and (c) $\lambda$ drawn from the uniform distribution on $[0,5]$.
• Figure 4: Left: A top-down surface plot of a solution to the NLSE \ref{['NLSE']} initialised with $u(x,0) = 2\,\mathrm{sech}(2 (x - x_0))$. Right: Model identification is performed on the tracked centre of mass of the soliton on the left (top) and numerical estimations of its second derivative (bottom).
• Figure 5: Relative error over $N\geq 200$ snapshots for different model identification algorithms to discover the harmonic oscillator $x" = -x$ using the centre of mass data shown in Figure \ref{['fig:NLSE']}. SINDy (yellow) and wtSINDy (green) methods use a sparsity parameter of $\eta = 10^{-2}$ (left) and $\eta = 10^{-4}$ (right).