mlsanityData to 7 Oct 2026

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics

arXiv 2302.13271 ↗First posted 26 Feb 2023cs.LG math.DS q-bio.QMcited by 20

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Abstract (arXiv)

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of C-infinity bump functions of varying support sizes. We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (https://github.com/MathBioCU/WENDy).

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 12 of its 74 references that are in the map. First posted on arXiv 26 Feb 2023.

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12 of 74 references.

References · 12

  1. 2020Weak SINDy For Partial Differential Equationscited by 154
  2. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 134
  3. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
  4. 2021Learning Mean-Field Equations from Particle Data Using WSINDycited by 29
  5. 2022Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimizationcited by 29
  6. 2020Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processescited by 18
  7. 2019ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systemscited by 17
  8. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
  9. 2022Online Weak-form Sparse Identification of Partial Differential Equationscited by 13
  10. 2022Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Populationcited by 11
  11. 2014Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditionscited by 5
  12. 2022A Joint Estimation Approach to Sparse Additive Ordinary Differential Equationscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 6; 2025: 6; 2026: 6.

20 papers.

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  1. 2024A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modelingcited by 22
  2. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  3. 2024Physics-informed active learning with simultaneous weak-form latent space dynamics identificationcited by 13
  4. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  5. 2024The Weak Form Is Stronger Than You Thinkcited by 11
  6. 2024Weak-Form Inference for Hybrid Dynamical Systems in Ecologycited by 7
  7. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  8. 2024Statistical Mechanics of Dynamical System Identificationcited by 5
  9. 2025Weak Form Scientific Machine Learning: Test Function Construction for System Identificationcited by 5
  10. 2025Learning Physically Interpretable Atmospheric Models from Data with WSINDycited by 4
  11. 2025Sparse Identification of Nonlinear Dynamics with Conformal Predictioncited by 3
  12. 2025Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movementcited by 3
8 more
  1. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
  2. 2024Sparse Regression for Discovery of Constitutive Models from Oscillatory Shear Measurementscited by 1
  3. 2025Overcoming error-in-variable problem in data-driven model discovery by orthogonal distance regressioncited by 1
  4. 2026Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identificationcited by 1
  5. 2026WGFINNs: Weak formulation-based GENERIC formalism informed neural networkscited by 0
  6. 2026Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamicscited by 0
  7. 2026Weak Dominant Balance for Robust Identification of Dynamically Consistent Fluid Flow Structurecited by 0
  8. 2026Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamicscited by 0