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This paper presents NeurEPDiff, a novel network to fast predict the geodesicsin deformation spaces generated by a well known Euler-Poincar\'e differentialequation (EPDiff). To achieve this, we develop a neural operator that for thefirst time learns the evolving trajectory of geodesic deformationsparameterized in the tangent space of diffeomorphisms(a.k.a velocity fields).In contrast to previous methods that purely fit the training images, ourproposed NeurEPDiff learns a nonlinear mapping function between thetime-dependent velocity fields. A composition of integral operators and smoothactivation functions is formulated in each layer of NeurEPDiff to effectivelyapproximate such mappings. The fact that NeurEPDiff is able to rapidly providethe numerical solution of EPDiff (given any initial condition) results in asignificantly reduced computational cost of geodesic shooting ofdiffeomorphisms in a high-dimensional image space. Additionally, the propertiesof discretiztion/resolution-invariant of NeurEPDiff make its performancegeneralizable to multiple image resolutions after being trained offline. Wedemonstrate the effectiveness of NeurEPDiff in registering two image datasets:2D synthetic data and 3D brain resonance imaging (MRI). The registrationaccuracy and computational efficiency are compared with the state-of-the-artdiffeomophic registration algorithms with geodesic shooting.

NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces