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. 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.

💡 Research Summary

The paper introduces a novel hybrid algorithm for simulating Willmore flow using a phase‑field formulation. Willmore flow is the L²‑gradient flow of the Willmore energy, i.e., the integral of squared mean curvature, and is governed by a fourth‑order nonlinear PDE that makes stable large‑time‑step discretizations difficult. Traditional approaches—finite‑element semi‑implicit schemes, level‑set methods, or fully implicit variational formulations—either require small time steps for stability or involve costly inner solves at each outer iteration.

The authors adopt the minimizing‑movement (time‑discrete variational) framework. For a time step τ they define an energy functional
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