images, registrations can be dramatically sped up, by either working with finite-dimensional Lie algebras [@zhang2015finite] and frequency diffeomorphisms [@zhang2017frequency], or by fast predictive image registration (FPIR) [@ref:yang2016; @quicksilver]. FPIR predicts the initial conditions (specifically, the initial momentum) of LDDMM, which fully characterize the geodesic and the spatial transformation using a *learned* a patch-based deep regression model. Because numerical optimization of standard LDDMM registration is replaced by a [*single*]{} prediction step, followed by optional correction steps [@quicksilver], FPIR is dramatically faster than optimization-based LDDMM without compromising registration accuracy, as measured on several registration benchmarks [@klein2009].

1ex Besides FPIR, other predictive image registration approaches have been proposed. Dosovitskiy et al. [@flownet2015] use a convolutional neural network (CNN) to directly predict optical flow. Liu et al. [@ref:liu] use an encoder-decoder network to synthesize video frames. Schuster et al. [@schuster2016optical] investigate strategies to improve optical flow prediction via a CNN. Cao et al. [@dinggang] use a sampling strategy and CNN regression to directly learn the mapping from moving and target image pairs to the final deformation field. Miao et al. [@miao2016real] use CNN regression for 2D/3D rigid registration. Sokooti et al.[@nonrigid] use CNNs to directly predict a 3D displacement vector field from input image 