STROBE Research Seminar

About this event

Title: Solve Less, Infer More: Weak-Form Methods for Equation Learning and Parameter Estimation

Presenters: Prof. Vanja Dukic and Prof. David Bortz, Department of Applied Mathematics, University of Colorado Boulder

Abstract: Weak-form scientific machine learning (WSciML) estimates the parameters, and even the form, of differential equations directly from noisy data, without repeatedly solving those equations. Integrating the governing equations against families of smooth test functions moves derivatives off noisy measurements and onto known functions. Many nonlinear inverse problems then become nearly linear regressions with principled, errors-in-variables uncertainty quantification. We introduce the framework (WSINDy for equation discovery [1,2] and WENDy for parameter estimation [3]), explain why it is robust to noise and fast, and survey applications in biology [4], nonlinear waves [5], and magnetohydrodynamics [6].

We then describe a simple, simulated toy problem of potential relevance to magnetic imaging. A magnetization field is reconstructed from limited-angle projections, with the reconstruction constrained by a micromagnetic equilibrium condition whose material parameters are learned from the data rather than assumed. We close with potential future directions for collaboration.

[1] D. A. Messenger and D. M. Bortz. Weak SINDy: Galerkin-based data-driven model selection. Multiscale Model. Simul., 19(3):1474–1497, 2021.

[2] D. A. Messenger and D. M. Bortz. Weak SINDy for partial differential equations. J. Comput. Phys., 443:110525, 2021.

[3] D. M. Bortz, D. A. Messenger, and V. Dukic. Direct estimation of parameters in ODE models using WENDy: Weak-form Estimation of Nonlinear Dynamics. Bull. Math. Biol., 85:110, 2023.

[4] R. Lyons, V. Dukic, and D. M. Bortz. Learning structured population models from data with WSINDy. PLOS Comput. Biol., 21(12):e1013742, 2025.

[5] S. Minor, V. Dukic, and D. M. Bortz. Learning effective soliton dynamics from scattering data. arXiv:2607.01545, 2026.

[6] G. Vasey, D. A. Messenger, D. M. Bortz, A. Christlieb, and B. O'Shea. Influence of initial conditions on data-driven model identification and information entropy for ideal MHD problems. J. Comput. Phys., 524:113719, 2025.