A hybrid minimizing movement and neural network approach to Willmore flow

TL;DR

This paper introduces a hybrid computational approach combining minimizing movement schemes with neural operators to efficiently simulate Willmore flow, demonstrating stability, accuracy, and applications in surface processing.

Contribution

It presents a novel hybrid method integrating neural approximations with optimization for simulating Willmore flow, improving efficiency and robustness.

Findings

Stable for large time steps

Reduced computational cost

Effective in surface fairing and shape reconstruction

Abstract

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. 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, the approach offers a robust and efficient tool for geometry processing.