Georgia Institute of Technology College of Sciences

Equation learning has been a holy grail of scientific research for decades. Only recently has the capability of learning equations directly from data become a computationally feasible task, due to the availability of high-resolution data and fast algorithms capable of surpassing the inherent combinatorial complexity of most model classes. Weak form equation learning has arisen as an advantageous framework for efficiently selecting models from data with noise and nonsmoothness, qualities inherent to observed data. By viewing the dynamics through the guise of test functions, the weak form affords a flexible representation of the governing equations that naturally incorporates these data maladies. More generally, the weak form has been shown to reveal alternative dynamical descriptions, such as coarse-grained and reduced-order models, opening the door to hierarchical model discovery. In this talk I will give a broad overview of historical advances in weak form equation learning and parameter inference, from the 1950s to WSINDy and more recent algorithms. I will then give an outlook for future research directions in this field, in light of now-known computational limitations and recently demonstrated successes, both theoretical and applied, with applications to molecular dynamics, plasma physics, cell biology, and weather forecasting.