The Weak Form Is Stronger Than You Think

TL;DR

This paper surveys the history and recent advances of the weak form in computational mathematics, highlighting its surprising robustness and efficiency in equation learning, parameter estimation, and coarse graining.

Contribution

It provides a comprehensive review of recent developments in weak form methods, emphasizing their enhanced noise robustness and computational advantages.

Findings

Weak form methods show increased noise robustness.

Recent advances improve accuracy and efficiency.

Software is available for reproducing results.

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 .