The Weak Form Is Stronger Than You Think
This is an incremental survey article for researchers in computational and applied mathematics, providing a detailed overview and references without introducing new methods.
The paper surveys the history and recent developments of the weak form in computational mathematics, highlighting advances in equation learning, parameter estimation, and coarse graining that offer improved noise robustness, accuracy, and computational efficiency.
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 .