WENDy: Weak-Form Nonlinear Dynamics Estimation

• WENDy is a weak-form framework that reformulates the estimation of nonlinear dynamics into an integral-based variational problem for improved noise robustness.
• It employs weighted sparse regression and quadrature-based discretization to mitigate noise amplification compared to traditional derivative-based methods.
• WENDy achieves high accuracy in parameter recovery for complex systems, demonstrating robust performance in ODE and PDE benchmarks such as logistic growth and reaction-diffusion models.

Weak form Estimation of Nonlinear Dynamics (WENDy) is a computational framework for data-driven identification and parameter estimation of nonlinear dynamical systems. Developed as a response to the limitations of derivative-based approaches such as SINDy, WENDy employs a variational (weak) formulation of dynamical equations, integrating against smooth test functions to fundamentally enhance noise robustness and numerical stability. The method encompasses sparse regression, errors-in-variables treatment, and efficient quadrature-based discretization, yielding interpretable and high-precision recovery of governing equations from noisy, potentially undersampled, or high-dimensional data.

1. Mathematical Formulation and Weak-Form Principle

At its core, WENDy recasts the identification or estimation of nonlinear ODEs and PDEs as a problem in the weak (integral) form. Rather than forming noisy pointwise derivatives from sampled data, the method proceeds by multiplying the model equation with smooth, compactly supported test functions and integrating over the domain, subsequently transferring all derivatives onto the test function via integration by parts. For scalar ODEs, this yields

$$-\int_{a}^{b} \varphi'(t) x(t) \, dt = \int_{a}^{b} \varphi(t) F(x(t)) \, dt,$$

where $\varphi$ is a test function, and $F$ is typically represented as a sparse expansion in a candidate library $\{f_j(x)\}$. With discretization on a temporal grid and application of quadrature (e.g., the trapezoidal rule), the relevant integrals become matrix–vector operations. For PDEs, the weak form is recast in terms of spatial-temporal test functions, leveraging separability and fast convolution, often computed via FFTs (Messenger et al., 2020, Messenger et al., 2020).

This weak formulation eliminates the need for pointwise finite differences, substantially attenuating amplification of high-frequency noise and leading to error bounds controlled by the test function smoothness and quadrature order (e.g., $\mathrm{O}(\Delta t^{p+1})$ for test functions with sufficient vanishing moments).

2. Algorithmic Structure and Sparse Regression

Identification proceeds as a weighted sparse regression problem. Defining test-function based integration matrices $V$, $V'$, and building the design (Gram) matrix $G = V \Theta(y)$ and right-hand side $b = -V' y$, WENDy formulates the inverse problem

$$\min_{w} (G w - b)^{\top} \Sigma^{-1} (G w - b) + \gamma^2 \| w \|_2^2,$$

where $\Sigma$ encodes the covariance structure induced by test function smoothing of measurement noise (Messenger et al., 2020, Bortz et al., 2023). Sparsity is enforced through iterative thresholding techniques, such as sequential thresholding least squares (STLS) or hard-thresholding $\ell_0$ procedures, yielding interpretable, low-dimensional models.

For systems nonlinear in parameters, WENDy defines an analytic weak-form negative log-likelihood, derives explicit expressions for the gradient and Hessian with respect to parameters, and leverages second-order non-convex optimization schemes (e.g., trust region or adaptive regularization) for maximum likelihood estimation (Rummel et al., 13 Feb 2025).

3. Noise Robustness and Theoretical Guarantees

The integration step acts as a low-pass filter, transforming the ill-posedness of numerical differentiation into a problem that averages measurement noise in a manner controlled by the choice and placement of test functions (Messenger et al., 2020, Messenger et al., 2022). The method's efficacy under white noise is rigorously established: provided the noise level $\sigma$ is below a critical threshold (scaling with sampling interval and test function order), coefficient recovery is asymptotically consistent. In the continuum limit (dense data), the parameter estimator’s error diminishes as $\|\hat\theta_n - \theta^*\| \leq C n^{-\alpha}$. Above the threshold, spurious model terms can appear, but denoising strategies or adaptive regularization restore reliable model recovery (Messenger et al., 2022).

Theoretical results quantify noise propagation through the weak form and its effect on statistical estimators, leading to principled design of covariance weighting in the regression and error bounds for identifiability assessment (Heitzman-Breen et al., 20 Jun 2025, Chawla et al., 3 Oct 2025).

4. Implementation Framework and Test Function Design

Test functions, typically smooth compactly supported “bump” functions or piecewise polynomials with vanishing derivatives up to prescribed order at boundaries, are central to the weak form. The support and order of these functions are chosen to maximize integration accuracy while adequately filtering out measurement noise. Automated selection of test function support based on data characteristics and error propagation analysis is recommended for optimal variance reduction (Messenger et al., 2020, Bortz et al., 2023).

Discretization employs quadrature matrices ($\Phi_k$, $\dot{\Phi}_k$), with corresponding assembly of the design matrix for regression. For large-scale applications (particularly spatiotemporal PDEs), convolutional evaluation and FFT-based implementations are utilized, resulting in algorithms with $\mathcal{O}(N^{D+1} \log N)$ computational complexity per regression step (Messenger et al., 2020).

WENDy is available as open-source MATLAB and Julia packages, with the latter providing efficient analytic differentiation for use in second-order optimization (Rummel et al., 13 Feb 2025, Bortz et al., 2023).

5. Performance, Applications, and Practical Advantages

WENDy achieves high accuracy and computational efficiency relative to output error least squares and other forward-solver–based methods. In canonical ODE benchmarks (Logistic, Lotka–Volterra, FitzHugh–Nagumo, Hindmarsh–Rose, Protein Transduction models, and others), it produces parameter estimates with lower bias, variance, and improved confidence interval coverage even at noise levels as high as 50%. It consistently outperforms SINDy and standard output-error regression in the presence of measurement noise or stiff dynamical regimes, with error reductions of 50%–90% and substantial speedups (typically orders of magnitude) (Bortz et al., 2023, Rummel et al., 13 Feb 2025, Chawla et al., 3 Oct 2025). Noise-robust estimation holds for both additive Gaussian and multiplicative log-normal noise models.

The method also enables accurate and interpretable sparse discovery for high-dimensional PDEs (inviscid/viscous Burgers’, KdV, Kuramoto–Sivashinsky, reaction–diffusion, Navier–Stokes), latent-space discovery in reduced-order models for complex systems, and design of control-oriented deep learning architectures for networked and multi-timescale systems (Messenger et al., 2020, Tran et al., 2023, Yu et al., 23 Jul 2024). In these contexts WENDy enables adaptive online estimation and robust latent variable identification under streaming or high-noise data (Messenger et al., 2022).

WENDy’s weak-form likelihood and covariance framework also enables direct quantification of parameter uncertainty, facilitating new identifiability analyses and rapid, robust assessment of practical identifiability via noise-aware mean-squared error criteria (Heitzman-Breen et al., 20 Jun 2025).

6. Limitations, Extensions, and Perspectives

While WENDy substantially mitigates noise amplification and aliasing, model selection can still be compromised if the noise level exceeds the weak-form averaging capacity (critical threshold), particularly for models with non-polynomial or oscillatory nonlinearities. Suitability and accuracy depend on careful test function selection and, in some regimes, on adaptive or multi-scale integration strategies. Extensions that have proven effective include: (i) integrating weak formulations with robust denoising, (ii) incorporating automatic differentiation for joint state and noise estimation (e.g., WmSINDy (López et al., 23 Oct 2024)), (iii) online or streaming regression approaches for time-varying systems (Messenger et al., 2022), and (iv) leveraging weak-form reductions for rapid identifiability and uncertainty quantification (Heitzman-Breen et al., 20 Jun 2025).

Practical aspects, such as integration of control signals and handling of unobserved or censored variables, remain active research directions. Computational bottlenecks in large systems are addressed through FFTs, blockwise or expansion training (for deep architectures), and efficient second-order solvers with analytic derivatives.

7. Summary Table: Key Features and Comparative Advantages

Feature	WENDy Weak-Form Estimation	Forward Error Least Squares	SINDy (Strong Form)
Numerical Differentiation	Avoided via integration	Required	Required
Noise Robustness	High	Low	Very Low (esp. at >1% noise)
Statistical Treatment	Weighted EIV, analytic cov.	Output mismatch	Plain least squares, lasso
Computational Efficiency	FFT/analytic derivatives	Many ODE solves	Moderate (if small data)
Applicability (ODEs/PDEs, noise)	ODE, PDE, high noise	ODE, PDE, low noise	ODE, low noise, PDE limited
Uncertainty Quantification	Yes (covariance, coverage)	Limited	Rarely explicit

WENDy’s weak-form approach, with its emphasis on integration for variance reduction and analytic propagation of noise to parameter estimates, achieves broad applicability and interpretability across the landscape of nonlinear dynamical system discovery and estimation. When paired with adaptive algorithms for test function and threshold optimization, WENDy constitutes a foundational tool for modern data-driven system identification in the presence of noise (Messenger et al., 2020, Bortz et al., 2023, Chawla et al., 3 Oct 2025).