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This paper investigates a long-standing question about the effect of surface roughness on turbulent flow: what is the equivalent roughness sand-grain height for a given roughness topography? Deep Neural Network (DNN) and Gaussian Process Regression (GPR) machine learning approaches are used to develop a high-fidelity prediction approach of the Nikuradse equivalent sand-grain height $k_s$ for turbulent flows over a wide variety of different rough surfaces. To this end, 45 surface geometries were generated and the flow over them simulated at $\hbox{Re}_τ=1000$ using direct numerical simulations. These surface geometries differed significantly in moments of surface height fluctuations, effective slope, average inclination, porosity and degree of randomness. Thirty of these surfaces were considered fully-rough and they were supplemented with experimental data for fully-rough flows over 15 more surfaces available from previous studies. The DNN and GPR methods predicted $k_s$ with an average error of less than 10% and a maximum error of less than 30%, which appears to be significantly more accurate than existing prediction formulas. They also identified the surface porosity and the effective slope of roughness in the spanwise direction as important factors in drag prediction.