A Generative Model-Free Form Deformation Approach for the Generation of Mesh Motions with Applications to PDE

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.