Research

 

 

Equation Discovery

Space-time plots of the Kuramoto-Sivashinsky equation: noisy data on top and the solution of the learned equation below

WSINDy (Weak-form Sparse Identification of Nonlinear Dynamics) identifies governing equations from noisy data by combining sparse regression with the weak form. Integration against smooth test functions replaces derivatives of the data, making the method robust to noise and efficient for both ODEs and PDEs.

Selected papers:

 

 

Parameter Estimation

Noisy FitzHugh-Nagumo time series data with the fitted model trajectories overlaid

WENDy (Weak-form Estimation of Nonlinear Dynamics) estimates parameters in differential equation models without repeatedly solving them. By pairing orthonormal test functions with an errors-in-variables statistical model, it is often faster and more accurate than solver-based approaches.

Selected papers:

  • Bortz, Messenger & Dukic, Direct estimation of parameters in ODE models using WENDy, Bull. Math. Biol. (2023). doi:10.1007/s11538-023-01208-6
  • Rummel, Messenger, Becker, Dukic & Bortz, WENDy for nonlinear-in-parameters ODEs, SIAM J. Sci. Comput. (to appear). arXiv:2502.08881
  • Messenger, Dwyer & Dukic, Weak-form inference for hybrid dynamical systems in ecology, J. R. Soc. Interface (2024). doi:10.1098/rsif.2024.0376

 

 

Uncertainty and Identifiability

Scatter plot of parameter estimates with nested confidence ellipses around the true value

Statistical properties of weak-form estimators, including bias, confidence interval coverage, and whether model parameters can be identified from the available data.

Selected papers:

  • Heitzman-Breen, Dukic & Bortz, A practical identifiability criterion leveraging weak-form parameter estimation, Bull. Math. Biol. (2026). doi:10.1007/s11538-026-01639-x
  • Chawla, Bortz & Dukic, Bias and coverage properties of the WENDy-IRLS algorithm, in Handbook of Visual, Experimental and Computational Mathematics (2026). doi:10.1007/978-3-032-16368-4_86

 

 

Test Functions and Theory

A noisy signal above families of smooth bump test functions with different support widths

Principled construction of test functions, and the conditions under which weak-form estimators are consistent.

Selected papers:

  • Messenger & Bortz, Asymptotic consistency of the WSINDy algorithm in the limit of continuum data, IMA J. Numer. Anal. (2025). doi:10.1093/imanum/drae086
  • Tran & Bortz, Weak form scientific machine learning: test function construction for system identification, SIAM J. Sci. Comput. (2026). doi:10.1137/25M1776020

 

 

Coarse Graining

A cloud of interacting particles shown over a smooth density estimate

Weak-form learning recovers effective models at coarser scales, such as mean-field equations from particle data and reduced Hamiltonian systems.

Selected papers:

 

 

Reduced-Order Modeling

Spiraling trajectories in a two-dimensional latent space, colored by a model parameter

Combining weak-form learning with latent space dynamics identification yields fast, noise-robust surrogate models of expensive simulations.

Selected papers:

  • Tran, He, Messenger, Choi & Bortz, Weak-form latent space dynamics identification, Comput. Methods Appl. Mech. Eng. (2024). doi:10.1016/j.cma.2024.116998
  • He, Tran, Bortz & Choi, Physics-informed active learning with simultaneous weak-form latent space dynamics identification, Int. J. Numer. Methods Eng. (2025). doi:10.1002/nme.7634

 

 

Control

The chaotic Lorenz attractor in gray with noisy observations, and a controlled trajectory in blue steered to a target equilibrium marked in gold

Models learned with WSINDy serve as the predictive model in model predictive control, with applications to fusion, drones, and chaotic systems.

Selected papers:

  • López, Partridge, De Pascuale, Lore, Christlieb, Becker & Bortz, WSINDy for model predictive control with applications to fusion, drones, and chaos, Proc. R. Soc. A (2026). doi:10.1098/rspa.2026.0413

 

 

Applications in Biology and Ecology

Log-scale population density over decades showing repeated boom-and-bust outbreaks

Applications include insect outbreaks and movement, cell interactions, structured populations, and dynamics on networks.

Selected papers:

  • Minor, Elderd, Van Allen, Bortz & Dukic, Weak-form learning for mean-field partial differential equations: an application to insect movement, SIAM J. Life Sci. (2026). doi:10.1137/25M1812038
  • Messenger, Dwyer & Dukic, Weak-form inference for hybrid dynamical systems in ecology, J. R. Soc. Interface (2024). doi:10.1098/rsif.2024.0376
  • Messenger, Wheeler, Liu & Bortz, Learning anisotropic interaction rules from individual trajectories in a heterogeneous cellular population, J. R. Soc. Interface (2022). doi:10.1098/rsif.2022.0412
  • Lyons, Dukic & Bortz, Learning structured population models from data with WSINDy, PLOS Comput. Biol. (2025). doi:10.1371/journal.pcbi.1013742
  • Tian, Messenger, Dukic, Rodríguez & Bortz, Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy, preprint (2026). arXiv:2605.30432

 

 

Applications in the Physical Sciences

A two-dimensional vorticity field with red and blue vortices and contour lines

Applications include atmospheric models learned from turbulent simulations and weather data, magnetized plasmas and MHD, and ultracold gases.

Selected papers:

 

A full list is on the Publications page.