Recent advances in data-driven modeling approaches have proven highly successful in a wide range of fields in science and engineering. In particular, learning governing equations via mimizing an equation error criteria, offers a powerful and explainable scientific machine-learning framework. However, the first generation of these methods has proven poorly suited to noise-corrupted data. In this talk, I will present our weak form approach and briefly discuss how it addresses several ubiquitous challenges within conventional model development, discretization, parameter inference, and model refinement. In particular, I will describe our equation learning (WSINDy) and parameter estimation (WENDy) algorithms. Our approach has exhibited surprising performance, accuracy, and robustness properties. In many applications, the method is an order of magitude more accurate, robust to orders of magnitude more noise, and multiple orders of magnitude faster than conventional approaches. I will demonstrate these performance properties via applications to several benchmark problems in ordinary, partial, and stochastic differential equations as well as coarse-graining and reduced order modeling.
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