We present predictions for the evolution of the galaxy dust-to-gas (DGR) and dust-to-metal (DTM) ratios from z=0 to 6, using a model for the production, growth, and destruction of dust grains implemented into the \simba\ cosmological hydrodynamic galaxy formation simulation. In our model, dust forms in stellar ejecta, grows by the accretion of metals, and is destroyed by thermal sputtering and supernovae. Our simulation reproduces the observed dust mass function at z=0, but modestly under-predicts the mass function by ~x3 at z ~ 1-2. The z=0 DGR vs metallicity relationship shows a tight positive correlation for star-forming galaxies, while it is uncorrelated for quenched systems. There is little evolution in the DGR-metallicity relationship between z=0-6. We use machine learning techniques to search for the galaxy physical properties that best correlate with the DGR and DTM. We find that the DGR is primarily correlated with the gas-phase metallicity, though correlations with the depletion timescale, stellar mass and gas fraction are non-negligible. We provide a crude fitting relationship for DGR and DTM vs. the gas-phase metallicity, along with a public code package that estimates the DGR and DTM given a set of galaxy physical properties.