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.

翻译：本文提出NeurEPDiff，一种能够快速预测由著名的欧拉-庞加莱微分方程（EPDiff）生成的变形空间中测地线的新型网络。为此，我们开发了一种神经算子，首次学习以微分同胚切空间（即速度场）参数化的测地变形演化轨迹。与纯粹拟合训练图像的先前方法不同，本文提出的NeurEPDiff学习时变速度场之间的非线性映射函数。通过在NeurEPDiff每层中构造积分算子与光滑激活函数的组合，可有效逼近此类映射。NeurEPDiff能够快速提供EPDiff的数值解（给定任意初始条件），从而显著降低高维图像空间中微分同胚测地线射击的计算成本。此外，NeurEPDiff的离散/分辨率不变特性使其在离线训练后，性能可推广至多种图像分辨率。我们通过在两个图像数据集（二维合成数据和三维脑部磁共振成像（MRI））上的配准实验验证了NeurEPDiff的有效性。将配准精度和计算效率与当前最先进的测地线射击微分同胚配准算法进行了比较。