Relative performance of COCA and COLA versus an emulator of the displacement field Ψ. We compute the summary statistics outlined in Sect. 3.5 for the final (a = 1) matter density field and compare results at both the training cosmology and a mis-specified one for COCA (COLA is evaluated at the correct cosmology in both cases). Although directly emulating Ψ produces a more accurate density field than simply emulating the momentum field p (with n_f = 0), using the COCA framework (emulating the frame of reference and employing additional force evaluations) yields the best performance. Moreover, even when using a mis-specified cosmology to predict the frame of reference, COCA significantly outperforms COLA for any given number of time steps.