The Weak Form Is Stronger Than You Think

TL;DR

This paper surveys the history and state-of-the-art of weak-form methods for learning governing equations, parameter estimation, and coarse-graining, emphasizing how test functions $\phi$ and integration-by-parts yield noise-robust, topology-aware representations. It highlights key methods such as WSINDy for robust PDE discovery and WENDy for fast, noise-tolerant parameter inference, illustrating strong performance on challenging problems like the Kuramoto–Sivashinsky PDE and latent-space dynamical systems. The authors discuss coarse-graining applications, including mean-field McKean–Vlasov PDE identification and homogenization of diffusive processes, and outline practical opportunities and theoretical questions for future work. Overall, the weak-form perspective provides a unifying, data-driven framework that enhances robustness, efficiency, and interpretability in learning dynamics across science and engineering.

Abstract

The weak form is a ubiquitous, well-studied, and widely-utilized mathematical tool in modern computational and applied mathematics. In this work we provide a survey of both the history and recent developments for several fields in which the weak form can play a critical role. In particular, we highlight several recent advances in weak form versions of equation learning, parameter estimation, and coarse graining, which offer surprising noise robustness, accuracy, and computational efficiency. We note that this manuscript is a companion piece to our October 2024 SIAM News article of the same name. Here we provide more detailed explanations of mathematical developments as well as a more complete list of references. Lastly, we note that the software with which to reproduce the results in this manuscript is also available on our group's GitHub website https://github.com/MathBioCU .

Figures (3)

• Figure 1: Schematic of weak-form PDE identification using the WSINDy_PDE algorithm. Solution data from the Kuramoto-Sivashinsky(KS) equation with 50% added noise is collected (z-axis limited to $[-10,10]$ for clarity). From noisy feature evaluations, a reference test function $\phi$ is identified to balance noise filtering and accuracy. Weak-form features are constructed using convolutions against $\phi$ and its derivatives. The governing equations approximately hold in this weak-form space, allowing accurate identification of model terms and coefficients.
• Figure 2: Equation Learning and Parameter Estimation. Left: For unlabeled particle trajectories (gray) in a multi-species population, force potentials are learned for each particle and then sorted into species (teal trajectories share a common learned model) (see MessengerWheelerLiuEtAl2022JRSocInterface). Center: Contour snapshot of the noisy measurements of the activator in a Reaction-Diffusion System with (inset) 5D ROM latent space (see TranHeMessengerEtAl2024ComputMethodsApplMechEng). Right: Comparison of parameter estimation performance on KS using equation error (EE) and output error (OE) methods (see BortzMessengerDukic2023BullMathBiol). Yellow circles represent WENDy, teal squares represent forward solver-NLS, and triangles are the corresponding geometric means.
• Figure 3: Coarse-graining. Left: Homogenization of a highly-oscillatory Fokker-Planck equation from particle data MessengerBortz2022PhysicaD. Right: reduction of (noisy) coupled charged particle motion (white) to coarse-grained Hamiltonian dynamics (teal) including inference of background electric potential $\widehat{V}_\mathbf{E}$ (contours). Particles start at the green markers and end at the red ones. Note the proximity of the full dynamics (circles) to the coarse-grained model (diamonds) MessengerBurbyBortz2024SciRep.