wtSINDy: Weighted Sparse ID of Nonlinear Dynamics

• wtSINDy is a data-driven method that refines traditional sparse identification by introducing non-uniform weights to enhance convergence in dynamic operator estimation.
• It mitigates edge effects and filters high-frequency noise, improving estimation accuracy across periodic, quasiperiodic, and stochastic regimes.
• The approach leverages weighted Birkhoff averages, clustering, and regularized regression techniques to achieve robust performance and reduced data requirements.

Weighted Extended Dynamic Mode Decomposition (wtEDMD) is a data-driven algorithm for extracting spectral properties, reduced-order models, and time-evolution operators associated with ergodic dynamical systems and stochastic differential equations. By introducing non-uniform, smoothly vanishing weights into the standard extended DMD framework, wtEDMD markedly accelerates convergence of finite-data estimates, especially in periodic and quasiperiodic regimes, and enhances filtering of high-frequency noise in stochastic systems. Rigorous convergence rates and practical algorithmic steps are detailed in the works of Bramburger, Colbrook, and Tahara et al. (Bou-Sakr-El-Tayar et al., 21 Nov 2025, Tahara et al., 26 Mar 2024).

1. Mathematical Foundations

In wtEDMD, core ergodic averages underpin operator estimation. For a measure-preserving, ergodic system $(X,T,\mu)$ and observable $g\in L^1(\mu)$, the classic Birkhoff average is

$$B_N(g)(x) = \frac{1}{N} \sum_{n=0}^{N-1} g(T^n(x)).$$

wtEDMD replaces the uniform weights by a smooth taper $w$ satisfying $w \in C^\infty([0,1])$, $\int_0^1 w = 1$, and all derivatives vanishing at endpoints,

$$WB_N(g)(x) = \frac{1}{\alpha_N} \sum_{n=0}^{N-1} w\left(\frac{n}{N}\right) g(T^n(x)), \quad \alpha_N = \sum_{n=0}^{N-1} w\left(\frac{n}{N}\right).$$

A canonical bump function is $w(x) = C \exp(-1/[x(1-x)])$, with normalization constant $C$ set by $\int_0^1 w=1$. These weights inherently suppress edge effects and facilitate super-polynomial or exponential convergence in non-chaotic settings (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

In the stochastic context (see (Tahara et al., 26 Mar 2024)), consider the Itô SDE

$dX_t = b(X_t)\,dt + \Sigma(X_t)\,dW_t,$

with drift $b$, diffusion matrix $\Sigma$, and associated generator

$$(\mathcal{L} f)(x) = \sum_{d=1}^D b_d(x)\,\partial_{x_d}f(x) + \tfrac{1}{2}\sum_{i,j=1}^D a_{ij}(x)\,\partial^2_{x_i x_j}f(x)$$

where $A(x) = \Sigma(x) \Sigma(x)^\top$. Locally weighted expectation operators are introduced to robustly estimate such derivatives,

$$\widetilde{b}(x) = \frac{\sum_{n=1}^{N-1} w_H(x,x_n) \frac{x_{n+1}-x_n}{\Delta t}}{\sum_{n=1}^{N-1} w_H(x,x_n)},$$

with Gaussian kernel $$w_H(x,x_n) = \exp(-\tfrac{1}{2} (x_n-x)^\top H^{-1} (x_n-x))$$.

2. wtEDMD Algorithm and Regression System

The standard EDMD constructs least-squares approximations of the Koopman operator using uniform averages over observables. wtEDMD modifies the approach by introducing a diagonal weight matrix $W$ composed of $w(n/N)$,

$W = \mathrm{diag}(w(0), w(1/N), \ldots, w(1)),$

and solves

$\min_K\|W^{1/2}\Phi - W^{1/2} \Psi K\|_F,$

yielding the optimal $K_w = (W^{1/2}\Psi)^\dagger (W^{1/2}\Phi)$ (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

Analogously, for the Koopman generator, the regression system is set up via weighted conditional moments at representative points,

$$L = \arg\min_{\widetilde L} \sum_{r=1}^M \| d\psi(x^{(r)}) - \widetilde L \psi(x^{(r)}) \|^2 + \lambda \sum_{i,j} |\widetilde \ell_{ij}|,$$

where $d\psi(x^{(r)})$ encodes generator derivatives estimated from weighted local statistics, and $L$ is the finite-dimensional Koopman generator (Tahara et al., 26 Mar 2024).

3. Convergence Properties

Weighted Birkhoff averages in wtEDMD exhibit markedly improved convergence rates compared to uniform averaging:

• Periodic dynamics: exponential decay in $N$.
• Quasiperiodic dynamics with smooth observables: super-polynomial error $\mathcal{O}(N^{-m})$ for all $m$.
• Analytic observables on quasiperiodic orbits: exponential convergence.
• Chaotic or stochastic settings: empirically at least $\mathcal{O}(1/N)$, matching conventional EDMD rates (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

These improvements are retained in finite-data projections of the Koopman operator and generator. Operator-norm bounds follow via weighted ergodic theory and convergence results for EDMD.

4. Practical Implementation and Clustering

In practice, wtEDMD requires careful selection of weights, basis size, and sample counts for stability:

• Typical weights: bump functions, or signal-processing windows (e.g., Tukey, Blackman–Harris) adjusted to kill endpoint derivatives.
• Sample count $N$: must exceed basis sizes $\max(L,R)$ for stable regression; in regular regimes, $N$ can often be reduced.
• Conditioning: monitor $Ψ^*WΨ$; if ill-conditioned, apply Tikhonov regularization or use stable SVD pseudoinverse.

For stochastic systems, Tahara et al. (Tahara et al., 26 Mar 2024) advocate a two-stage clustering methodology:

1. Outlier Removal: IsolationForest eliminates anomalous trajectory points.
2. Representative Selection: k-means yields $M$ centroids to focus the regression.
3. Local Structure Capture: Dirichlet-process Gaussian-mixture model (DPMM) clusters the dataset, with each centroid inheriting its cluster covariance for anisotropic weighting.

This enables local moments and generator derivatives to reflect trajectory-dependent noise geometry, yielding substantially better drift/diffusion estimates and Koopman generator matrices.

5. Numerical Illustrations

To clarify performance gains and practical trade-offs, select benchmarks are summarized in the table below.

Example & Regime	EDMD Error	wtEDMD Error
Standard map, quasiperiodic ($\lambda=0.25$), 9×9 Fourier	$\sim 10^{-1}$ ($N=10^3$)	$\sim 10^{-4}$, plateaus at precision
Lid-driven cavity, autocorrelation (periodic)	$10^{-2}$ ($N=1100$)	$10^{-6}$ ($M=1000$, weighted)
El Niño diffusion forecast, RMSE/correlation (16 mo lead)	Baseline	wtEDMD: correlation gain ∼0.1

A plausible implication is that, in non-chaotic settings, wtEDMD can reduce data requirements and improve estimation accuracy by several orders of magnitude. For chaotic systems, wtEDMD matches EDMD rates and does not introduce slowdown.

6. Limitations and Best Practices

wtEDMD's efficacy is regime-dependent. In chaotic or strongly stochastic dynamics, speed-up is limited to $\mathcal{O}(1/N)$. Noisy or insufficiently smooth observables can be amplified at endpoints if heavy tapering is applied—moderate windows or pre-filtering may be preferred. Overcomplete basis sets necessitate significantly larger data sample sizes to avoid ill-conditioning in the weighted pseudocovariance matrices.

Weight profile tuning balances edge suppression and sample efficiency: strong damping loses sample size; 10–20% Tukey windows provide a compromise. Always use robust SVD-based pseudoinverses and regularize Gram matrices when required.

7. Contextual Significance

wtEDMD represents a refinement of time-averaged data-driven operator estimation found in DMD, EDMD, SINDy, spectral measure estimation, and diffusion forecasting. By integrating weighted Birkhoff theory and local weighted conditional expectation (in the stochastic generator context), the method directly addresses finite-data limitations and improves convergence without additional coding overhead. Its two-stage clustering for stochastic systems reflects recent advances in capturing anisotropic noise via unsupervised learning priors, contributing to improved generator estimation in nonlinear systems (Bou-Sakr-El-Tayar et al., 21 Nov 2025, Tahara et al., 26 Mar 2024).