Physics-informed learning under mixing: How physical knowledge speeds up learning

Physics-informed machine learning encompasses a wide taxonomy of approaches that combine physical knowledge and learning algorithms to address two main tasks: (i) enhancing physical models (given, e.g., by systems of partial differential equations) through data-driven methods to improve their accuracy and numerical solvability; (ii) improve the learning algorithms' performance by including physical information, e.g., as additional constraint (Karniadakis et al., 2021; Meng et al., 2025). Focusing on the second class of methods, surveyed in Rai and Sahu, 2020; von Rueden, Mayer, et al., 2023, the resulting approaches turn out to be practically effective in terms of data efficiency, generalization capability and interpretability, especially in view of downstream tasks such as safe learning-based control (Nghiem et al., 2023; Drgona et al., 2025). However, theoretically quantifying the beneficial impact of physical information into learning algorithms is challenging and still an active research question (see von Rueden, Garcke, and Bauckhage, 2023 and references therein).