WENDy-IRLS ODE Parameter Estimation

• The paper introduces a parameter estimation method for nonlinear ODEs that reformulates the problem using a weak-form integral approach coupled with iteratively reweighted least squares.
• The paper demonstrates robust statistical performance across various benchmark models, addressing errors-in-variables with quantifiable bias and coverage under multiple noise types.
• The paper provides a comprehensive bias and coverage analysis, highlighting the algorithm’s efficiency and limitations under high noise and nonlinear dynamics.

The WENDy-IRLS algorithm is a parameter estimation method for nonlinear differential equations (ODEs) that utilizes a weak-form integral approach and iteratively reweighted least squares (IRLS) to achieve high robustness to noise and computational efficiency. By reformulating parameter estimation into a weak (integral) formulation and addressing the errors-in-variables (EiV) challenge through a statistically principled IRLS update, the algorithm systematically delivers estimates with quantifiable bias and coverage properties. Comprehensive benchmarking against ODE models such as Logistic, Lotka–Volterra, FitzHugh–Nagumo, Hindmarsh–Rose, and the Protein Transduction Benchmark (PTB) provides insight into the estimator's statistical behavior across a broad range of noise types and strengths (Chawla et al., 3 Oct 2025).

1. Mathematical Framework

WENDy-IRLS begins by expressing the ODE system in feature-expanded form: $\frac{d\mathbf{u}}{dt} = \Theta(\mathbf{u}) W$ where $\mathbf{u}(t)\in\mathbb{R}^d$, $\Theta(\mathbf{u})$ denotes a collection of nonlinear features (e.g., monomials, cross-terms), and $W$ is the parameter matrix. This ODE is converted to its weak form by multiplying both sides by a smooth test function $\varphi(t)$ with compact support and integrating by parts: $$-\int_0^T \mathbf{u}(t)\varphi'(t)dt = \int_0^T \varphi(t)\Theta(\mathbf{u}(t))dt \cdot W$$ Measurement data $\mathbf{U}$, sampled at $M+1$ time points, are introduced via quadrature to estimate the integrals, yielding the discretized regression system: $\min_{W} \|\mathrm{vec}(G W - B)\|_2^2$ where $G = \Phi Q \Theta(\mathbf{U})$ (with $\Phi$ the test function evaluation matrix, $Q$ the quadrature matrix), and $B = -\dot{\Phi} Q \mathbf{U}$.

Due to the fact that both $G$ and $B$ are computed from noisy data $\mathbf{U}$, the problem becomes an EiV regression, with parameter-dependent residual covariance.

2. IRLS Update and Statistical Correction for Errors-in-Variables

Classical least squares fails in the EiV setting because standard assumptions about independent residuals are violated. WENDy-IRLS addresses this by estimating, at each iteration $n$, a covariance matrix $C^{(n)}$ for the regression residuals. Each parameter update is then computed via weighted least squares: $$W^{(n+1)} = (G^\top [C^{(n)}]^{-1} G)^{-1} G^\top [C^{(n)}]^{-1} B$$ where $C^{(n)}$ is calculated using a first-order Taylor expansion of the residual with respect to noise, incorporating both the test function structure and the current parameter estimate.

The iteration continues until a user-defined convergence criterion is satisfied.

3. Bias and Coverage Analysis

Two critical properties are investigated:

• Coverage: The empirical probability that a confidence interval for parameter $w_i$ contains the true value $w_i^*$. The estimator’s asymptotic normality implies (in the absence of strong nonlinearities and under typical noise regimes) that $w_i \pm 1.96 \hat{\sigma}_{w_i}$ achieves nominal 95% coverage.
• Bias: The average difference $E(w_i^* - w_i)$ and relative bias $E(w_i^* - w_i)/w_i^*$ quantify systematic deviation for each parameter. These metrics are evaluated under increasing noise and varying noise distributions for each ODE system.

4. Performance Across Noise Distributions

The algorithm is benchmarked under:

• Additive Normal noise (homoscedastic, symmetric)
• Additive Censored Normal (ACN) noise (normal noise, censored to preserve positivity)
• Additive Truncated Normal (ATN) noise (normal noise, truncated below to maintain positivity, introducing skewness)
• Multiplicative Log-Normal (MLN) noise (heteroscedastic, state-proportional variance)

Key findings include:

• Normal Noise: For parameters multiplying linear terms (e.g., $w_1$), estimators are effectively unbiased and coverage remains at ~95% until noise exceeds 70% of typical state magnitudes (e.g., Logistic model). For parameters multiplying nonlinear terms (e.g., quadratic interactions $w_2$), increased noise leads to decreasing coverage and bias away from nominal.
• ACN & ATN Noise: Coverage often remains close to normal up to moderate noise, but skewness (ATN) introduces bias and, at high noise, bimodal estimator distributions.
• MLN Noise: Parameter and coverage stability decline as heteroscedasticity increases, particularly for those parameters most sensitive to state amplitude. Some parameters maintain near-nominal coverage substantially above 70% noise, but relative bias increases with noise (especially for those modulating higher-order or nonlinear terms).

These results systematically characterize the estimator's robustness and limits.

5. Evaluation on Benchmark ODE Systems

The estimator’s statistical properties are studied for:

• Logistic growth: $du/dt = w_1 u + w_2 u^2$. $w_1$ (linear) is consistently well-estimated. $w_2$ (quadratic) is more sensitive, with bias and sub-nominal coverage at high noise or under heteroscedasticity.
• Lotka–Volterra: Predator-prey with multilinear terms. Parameters multiplying nonlinearities exhibit greater bias/variance.
• FitzHugh–Nagumo, Hindmarsh–Rose: Chaotic and stiff neuronal systems. These require higher data resolution and lower noise to attain nominal coverage for parameters, especially those linked to higher-order terms.
• Protein Transduction Benchmark (PTB): Multi-compartment system dominated by (piecewise) linear and Michaelis–Menten terms. WENDy-IRLS achieves low bias and good coverage for most parameters except those governing the most nonlinear interactions, where more data or refined normalization may be necessary.

6. Applications and Limitations

WENDy-IRLS is particularly well-suited for systems biology, ecology, and engineering domains where:

• Direct ODE integration is computationally intensive.
• High noise or heteroscedasticity is present in measured time-series.
• A solver-free, statistically principled, and computationally efficient inference method is required.

Its strength lies in:

• Explicitly correcting for errors-in-variables via IRLS.
• Surpassing traditional finite-difference or sparse symbolic regression methods in both speed and robustness, especially in high-noise regimes.
• Avoiding artifacts associated with numerical differentiation or black-box regression.

Identified limitations include:

• Nonlinear and high-order terms induce estimator bias and reduced coverage at high noise and low data resolution.
• For skewed or censored noise models, some parameters may require bias-correction or alternative confidence interval metrics.
• For highly nonlinear systems, sufficient temporal resolution and/or alternative test functions may be needed to ensure estimators maintain coverage near nominal.

7. Summary Table: Performance Across Models and Noise Types

Model	Linear Param. Coverage	Nonlinear Param. Coverage	Stable to MLN Noise	Sensitive to Data Resolution
Logistic	High	Drops at high noise	Moderate	No
Lotka–Volterra	High/moderate	Drops at high noise	Moderate	No
FitzHugh-Nagumo, HR	Moderate	Low at high noise	Low	Yes
Protein Trans. Bench.	High	Moderate	High (linear terms)	Moderate

The above illustrates that linear parameters are generally robust, while nonlinear/high-order parameters require careful data quality and model consideration for unbiased estimation and accurate confidence intervals.

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In sum, the WENDy-IRLS algorithm operationalizes weak-form ODE regression using an IRLS update specifically tailored to the errors-in-variables setting. This ensures high computational efficiency, statistical tractability, and practical applicability across a range of differential equation models and noise regimes, with a comprehensive understanding of potential bias and coverage limitations (Chawla et al., 3 Oct 2025).