Information-Theoretic Buckingham-Pi Theorem: Application to Wall Model Discovery for Compressible Flow Over Roughness

Physical laws and models must rely on dimensionless variables. The Buckingham Pi theorem offers a systematic approach for obtaining dimensionless numbers for a given problem. However, these dimensionless numbers are not unique, as there is an infinite set of valid solutions. We introduce an information-theoretic, data-driven dimensional analysis method that identifies the best-performing dimensionless inputs to predict the non-dimensional quantity of interest. The approach is grounded in the information-theoretic bounds to the irreducible model error, which guarantees that the inputs identified maximize the predictability of the output regardless of the chosen modeling approach. The method involves the maximization of the mutual information between inputs and output, which is efficiently solved using the covariance matrix adaptation evolution strategy algorithm. We first validate the Information-Theoretic Buckingham Pi Theorem with a synthetic dataset with known scaling laws. Then, the dimensional learning is applied to discover a wall model for compressible flows over rough walls using a recent DNS database. The best dimensionless inputs and outputs for the wall model are utilized to train an artificial neural network model able to predict wall shear stress and heat flux within 10% accuracy.