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We present MeshODE, a scalable and robust framework for pairwise CAD model deformation without prespecified correspondences. Given a pair of shapes, our framework provides a novel shape feature-preserving mapping function that continuously deforms one model to the other by minimizing fitting and rigidity losses based on the non-rigid iterative-closest-point (ICP) algorithm. We address two challenges in this problem, namely the design of a powerful deformation function and obtaining a feature-preserving CAD deformation. While traditional deformation directly optimizes for the coordinates of the mesh vertices or the vertices of a control cage, we introduce a deep bijective mapping that utilizes a flow model parameterized as a neural network. Our function has the capacity to handle complex deformations, produces deformations that are guaranteed free of self-intersections, and requires low rigidity constraining for geometry preservation, which leads to a better fitting quality compared with existing methods. It additionally enables continuous deformation between two arbitrary shapes without supervision for intermediate shapes. Furthermore, we propose a robust preprocessing pipeline for raw CAD meshes using feature-aware subdivision and a uniform graph template representation to address artifacts in raw CAD models including self-intersections, irregular triangles, topologically disconnected components, non-manifold edges, and nonuniformly distributed vertices. This facilitates a fast deformation optimization process that preserves global and local details. Our code is publicly available.