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While physical phenomena are stated in terms of physical laws that arehomogeneous in all dimensions, the mechanisms and patterns of the physicalphenomena are independent of the form of the units describing the physicalprocess. Accordingly, across different conditions, the similarity of a processmay be captured through a dimensionless reformulation of the physical problemwith Buckingham $\Pi$ theorem. Here, we apply Buckingham $\Pi$ theorem forcreating dimensionless indices for capturing the similarity of the floodprocess, and in turn, these indices allow machine learning to map thelikelihood of pluvial (flash) flooding over a landscape. In particular, we usethese dimensionless predictors with a logistic regression machine learning (ML)model for a probabilistic determination of flood risk. The logistic regressionderived flood maps compare well to 2D hydraulic model results that are thebasis of the Federal Emergency Management Agency (FEMA) maps. As a result, theindices and logistic regression also provide the potential to expand existingFEMA maps to new (unmapped) areas and a wider spectrum of flood flows andprecipitation events. Our results demonstrate that the new dimensionlessindices capture the similarity of the flood process across differenttopographies and climate regions. Consequently, these dimensionless indices mayexpand observations of flooding (e.g., satellite) to the risk of flooding innew areas, as well as provide a basis for the rapid, real-time estimation offlood risk on a worldwide scale.

Pi theorem formulation of flood mapping