Schematic illustration of the (a) COLA and (b) COCA formalism for cosmological simulations. In COLA, we solve the equations of motion in the frame of reference given by LPT, so we compute the residual (‘res’) between the LPT trajectory and the true position, x, of particles. In COCA, we emulate a frame of reference closer to the true trajectory by adding a ML contribution to LPT. Thus, we can solve for the (smaller) residuals between x and the emulated frame.