A Generative Model-Free Form Deformation Approach for the Generation of Mesh Motions with Applications to PDE
arXiv:2607.13202
The paper presents a topology‑agnostic framework that models 3D shape deformations as an ODE flow parameterized by a time‑dependent free‑form deformation lattice, and couples it with a flow‑based generative model (TarFlow) to learn compact latent dynamics for efficient sampling, optimization, and reduced‑order modeling of mesh motions in PDE applications.
Abstract
We introduce a topology-agnostic framework for matching deformations of three-dimensional shapes with non-isomorphic mesh graphs by modelling the deformation as the flow of an Ordinary Differential Equation (ODE). The velocity field is parameterised by a time-dependent Free Form Deformation (FFD), expressed through displacements of a coarse control lattice, yielding a smooth and low-dimensional representation that decouples the deformation model from the discretisation of the source and target surfaces. Under mild regularity assumptions, we prove that the induced ODE map is a universal approximator (in the sup norm) for mappings between genus-0 surfaces, providing a theoretical expressivity guarantee. To further compress the representation and enable probabilistic inference, we couple the ODE--FFD model with a flow-based generative approach in the TarFlow framework, learning a compact latent parametrisation over time series of FFD maps. The resulting method supports efficient sampling and optimisation of plausible deformation trajectories while preserving mesh quality, and it enables scalable reduced-order modelling. Experiments on deforming-body flow benchmarks demonstrate improved accuracy and computational efficiency of reduced-order models constructed from the learned latent dynamics.
48 pages, 14 figures