Role of physics in physics-informed machine learning

Physical systems are characterized by inherent symmetries, one of which is encapsulated in the units 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, machine learning (ML) strategies for the discovery of constitutive laws seldom subject experimental and/or numerical data to dimensional analysis. We demonstrate the potential of dimensional analysis to significantly enhance the interpretability and generalizability of ML-discovered secondary laws. Our numerical experiments with creeping fluid flow past solid ellipsoids show how dimensional analysis enables both deep neural networks and sparse regression to reproduce old results, e.g., Stokes law for a 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 invariances into ML-based techniques for equation discovery.