A hybrid minimizing movement and neural network approach to Willmore flow

We present a hybrid method combining a minimizing movement scheme with neural operators for the simulation of phase field-based Willmore flow. The minimizing movement component is based on a standard optimization problem on a regular grid, whereas the functional to be minimized involves a neural approximation of mean curvature flow proposed by Bretin et al. [J. Comput. Phys. 470 (2022), article no. 111579]. Numerical experiments confirm stability for large time step sizes, consistency, and significantly reduced computational cost compared to a traditional finite element method. Moreover, applications demonstrate its effectiveness in surface fairing and reconstructing of damaged shapes. Thus, our approach offers a robust and efficient tool for geometry processing.