that FPIR can predict deformation fields similar to the ones obtained using optimization-based LDDMM, even for the subtle changes seen in longitudinal imaging data. However, it remains to be seen how a predictive model performs for image regression. Fig. \[fig:example\] shows an exemplary regression result. In this specific case, large changes can be observed around the ventricles. To illustrate differences between the methods, Fig. \[fig:example\] shows regression results based on optimization-based LDDMM, for FPSGR *without* a correction network, and for FPSGR *with* a correction network. All three methods successfully capture the expanding ventricles and generally capture the image changes. Both FPSGR methods show results that are highly similar to SGR using optimization-based LDDMM. Hence, FPSGR is useful for longitudinal image regression. To further quantify the regression accuracy, we compute the overlay error between measured images and the images on the geodesic as $$\begin{aligned}
\label{eq:overlay}
E_{overlay} (I_0 \circ \Phi^{-1}_{t_i}, Y_{i}) = \frac{1}{|\Omega|} \lVert I_0 \circ \Phi^{-1}_{t_i} - Y_{i}\rVert_{L_1}\end{aligned}$$ where $\Omega$ is the brain area, $I_0 \circ \Phi^{-1}_{t_i}$ is the regressed image at time $t_i$ and $Y_{i}$ is the measured image at time $t_i$. Table \[tab:overlay\] shows the overlay error for the population of 100 subjects which includes all diagnostic groups in 