Physical systems are characterized by inherent symmetries, one of which is encapsulated in theunits of their parameters and system states. These symmetries enable a lossless order-reduction, e.g.,via dimensional analysis based on the Buckingham theorem. Despite the latter's benefits, machinelearning (ML) strategies for the discovery of constitutive laws seldom subject experimental and/ornumerical data to dimensional analysis. We demonstrate the potential of dimensional analysis to significantlyenhance the interpretability and generalizability of ML-discovered secondary laws. Ournumerical experiments with creeping fluid flow past solid ellipsoids show how dimensional analysisenables both deep neural networks and sparse regression to reproduce old results, e.g., Stokes law fora sphere, and generate new ones, e.g., an expression for an ellipsoid misaligned with the flow direction.Our results suggest the need to incorporate other physics-based symmetries and invariancesinto ML-based techniques for equation discovery.
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Physics Inventory of Quantitative Literacy: A tool for assessing mathematical reasoning in introductory physics
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Quantitative reasoning is an essential learning objective of physics instruction. The Physics Inventory for Quantitative Literacy (PIQL) is a published assessment tool that has been developed for calculus-based physics courses to help instructors evaluate whether their students learn to reason this way. However, the PIQL is not appropriate for the large population of students taking physics who are not enrolled in, or have not completed, calculus. To address this need, we have developed the General Equation-based Reasoning inventory of QuaNtity (GERQN). The GERQN is an of the PIQL and is appropriate for most physics students; the only requirement is that students have taken algebra, so they are familiar with the use of variables, negative quantities, and linear functions. In this paper, we present the development and validation of the GERQN, and a short discussion on how the GERQN can be used by instructors to help their students learn.more » « less
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