Weak Form Scientific Machine Learning: Test Function Construction for System Identification

Abstract. Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we provide a mathematical motivation for a novel data-driven approach to constructing test functions for optimal weak form system identification. Specifically, we introduce a Single-scale-Local strategy for constructing the test function set. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of both minimal parameter estimation and forward simulation error. We further compare the proposed approach with previously introduced strategies for constructing test function sets, including the orthogonal Multiscale Global approach [Bortz, Messenger, and Dukic, Bull. Math. Biol., 85 (2023), 110] as well as constructions based on Spectral Matching [Messenger and Bortz, J. Comput. Phys., 443 (2021), 110525], demonstrating improved accuracy in parameter recovery and forward simulation, as well as enhanced computational efficiency. Code implementing the proposed method is available at https://github.com/MathBioCU/pyWENDy .