  This paper presents NeurEPDiff, a novel network to fast predict the geodesics
in deformation spaces generated by a well known Euler-Poincar\'e differential
equation (EPDiff). To achieve this, we develop a neural operator that for the
first time learns the evolving trajectory of geodesic deformations
parameterized in the tangent space of diffeomorphisms(a.k.a velocity fields).
In contrast to previous methods that purely fit the training images, our
proposed NeurEPDiff learns a nonlinear mapping function between the
time-dependent velocity fields. A composition of integral operators and smooth
activation functions is formulated in each layer of NeurEPDiff to effectively
approximate such mappings. The fact that NeurEPDiff is able to rapidly provide
the numerical solution of EPDiff (given any initial condition) results in a
significantly reduced computational cost of geodesic shooting of
diffeomorphisms in a high-dimensional image space. Additionally, the properties
of discretiztion/resolution-invariant of NeurEPDiff make its performance
generalizable to multiple image resolutions after being trained offline. We
demonstrate the effectiveness of NeurEPDiff in registering two image datasets:
2D synthetic data and 3D brain resonance imaging (MRI). The registration
accuracy and computational efficiency are compared with the state-of-the-art
diffeomophic registration algorithms with geodesic shooting.
