A unified sparse optimization framework to learn parsimonious physics-informed models from data

Machine learning (ML) is redefining what is possible in data-intensive fields\nof science and engineering. However, applying ML to problems in the physical\nsciences comes with a unique set of challenges: scientists want physically\ninterpretable models that can (i) generalize to predict previously unobserved\nbehaviors, (ii) provide effective forecasting predictions (extrapolation), and\n(iii) be certifiable. Autonomous systems will necessarily interact with\nchanging and uncertain environments, motivating the need for models that can\naccurately extrapolate based on physical principles (e.g. Newton's universal\nsecond law for classical mechanics, $F=ma$). Standard ML approaches have shown\nimpressive performance for predicting dynamics in an interpolatory regime, but\nthe resulting models often lack interpretability and fail to generalize. We\nintroduce a unified sparse optimization framework that learns governing\ndynamical systems models from data, selecting relevant terms in the dynamics\nfrom a library of possible functions. The resulting models are parsimonious,\nhave physical interpretations, and can generalize to new parameter regimes. Our\nframework allows the use of non-convex sparsity promoting regularization\nfunctions and can be adapted to address key challenges in scientific problems\nand data sets, including outliers, parametric dependencies, and physical\nconstraints. We show that the approach discovers parsimonious dynamical models\non several example systems. This flexible approach can be tailored to the\nunique challenges associated with a wide range of applications and data sets,\nproviding a powerful ML-based framework for learning governing models for\nphysical systems from data.\n