Discovering the governing laws underpinning physical and chemical phenomena entirely from data is a key step towards understanding and ultimately controlling systems in science and engineering. Noisy measurements and complex, highly nonlinear underlying dynamics hinder the identification of such governing laws. In this work, we introduce a machine learning framework rooted in moving horizon nonlinear optimization for identifying governing equations in the form of ordinary differential equations from noisy experimental data sets. Our approach evaluates sequential subsets of measurement data, and exploits statistical arguments to learn truly parsimonious governing equations from a large dictionary of basis functions. The proposed framework reduces gradient approximation errors by implicitly embedding an advanced numerical discretization scheme, which improves robustness to noise as well as to model stiffness. Canonical nonlinear dynamical system examples are used to demonstrate that our approach can accurately recover parsimonious governing laws under increasing levels of measurement noise, and outperform state of the art frameworks in the literature. Further, we consider a non-isothermal chemical reactor example to demonstrate that the proposed framework can cope with basis functions that have nonlinear (unknown) parameterizations.
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