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A hybrid minimizing movement and neural network approach to Willmore flow

Martin Rumpf, Josua Sassen, Christoph Smoch

TL;DR

Willmore flow is the $L^2$-gradient flow of the Willmore energy $w[x]=\frac{1}{2}\int_\Gamma |\mathbf{h}|^2\,d\mathcal{H}^{d-1}$, and poses a fourth-order PDE challenging for stable, large-time-step simulations. The paper proposes a hybrid method that combines a phase-field minimizing-movement time discretization with a neural-operator approximation of the inner mean-curvature step, using $v^{f,\kappa}_{\tilde{\tau}}[u]=f_{\tilde{\tau}}(\kappa_{\tilde{\tau}}*u)$. The approach yields a time-stepping scheme stable for large time steps and reduces computational cost relative to traditional finite element approaches, while enabling surface fairing and damaged-shape reconstruction. The results underscore the viability of neural operators within PDE-constrained variational schemes for geometry processing and point toward learning direct flow operators for Willmore flow in future work.

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.

A hybrid minimizing movement and neural network approach to Willmore flow

TL;DR

Willmore flow is the -gradient flow of the Willmore energy , and poses a fourth-order PDE challenging for stable, large-time-step simulations. The paper proposes a hybrid method that combines a phase-field minimizing-movement time discretization with a neural-operator approximation of the inner mean-curvature step, using . The approach yields a time-stepping scheme stable for large time steps and reduces computational cost relative to traditional finite element approaches, while enabling surface fairing and damaged-shape reconstruction. The results underscore the viability of neural operators within PDE-constrained variational schemes for geometry processing and point toward learning direct flow operators for Willmore flow in future work.

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.

Paper Structure

This paper contains 10 sections, 1 theorem, 23 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Synthesis of the time discrete Willmore flow
  3. Phase field based minimizing movement scheme for Willmore flow.
  4. Convolution based approximation of mean curvature.
  5. Spatial discretization
  6. Minimizing movement scheme for Willmore flow.
  7. Neural approximation of discrete mean curvature flow.
  8. Numerical experiments
  9. Applications in image and geometry processing
  10. Conclusions

Key Result

Proposition 2.4

For $\Omega=(0,1)^d$, $u^0\in L^2(\Omega)$ periodically extended on $\mathbb{R}^d$, and $e^\varepsilon[u^0,\cdot]$ as in eq:discrete_willmore_flow_energy with $v_{\tilde{\tau}}$ of the form $v_{\tilde{\tau}}[u] = f(\kappa\ast u)$ for $f\in C(\mathbb{R})$ and $\kappa \in L^2(\mathbb{R}^d)$ there exis

Figures (9)

  • Figure 1: Learned networks for $\varepsilon = 2^{-6}$ and $\tilde{\tau} = 2^{-14}$ are displayed via a color coding of the learned kernels $K$ and the graphs of learned activation function $F^\theta$ on the interval $[-1, 1]$ for increasing $n = 128,\, 256,\, 512$, with stencil widths $n_K=17,\,33,\,65$, respectively.
  • Figure 2: Convergence validation for mean curvature flow with fixed $\varepsilon = 2^{-6}$ and $\tilde{\tau} = 2^{-14}$ while increasing $n$, with $n_K = \tfrac{n}{8} + 1$. We plot the average $L^2$-error to the analytic solution along time for 30 circles with radii $r_i = 0.05\pi + \tfrac{0.15\pi i}{30}\,$, $i=0,\dots,29$. The line-styles correspond to the different methods and the colors to the varying resolution. For comparison: for the averaged $L^2$ distance between the solution at time $0.004$ and at the initial time zero, one obtains $\tfrac{1}{30}\sum_{i=1}^{30} \|U_{R(r_i, 0.004)} - U_{R(r_i, 0)}\|_{L^2} \approx 0.133$.
  • Figure 3: Convergence tests for Willmore flow with fixed $\varepsilon = 2^{-6}$ and $\tilde{\tau} = 2^{-14}$ while increasing $n$, with $n_K = \tfrac{n}{8} + 1$. As in \ref{['fig:ConvergenceTestMCF']}, the line-styles correspond to the different methods for approximating the mean curvature evolution and the colors to the varying resolution. In (a), we plot the average $L^2$-error to the analytic solution for 30 circles with radii $r_i = 0.05\pi + \tfrac{0.15\pi i}{30}\,$, $i=0,\dots,29$ over time. For comparison: for the averaged $L^2$ distance between the solution at time $0.004$ and at the initial time zero, one obtains$\tfrac{1}{30}\sum_{i=1}^{30} \|U_{R_W(r_i, 0.004)}- U_{R_W(r_i, 0)}\|_{L^2} \approx 0.414$. In (b), we plot the $L^2$-error of the evolution of a rectangle sized $0.4\times 0.2$ compared to a fine implicit finite element solution ($n=2048$).
  • Figure 4: Willmore flow of a rectangle with side lengths $0.4\times 0.2$ using $\varepsilon=2^{-6}$. Results of our hybrid scheme on resolution $n=256$ (top row), fully implicit finite element solution of the nested scheme on resolution $n=256$ (middle row) and on resolution $n=2048$ (bottom row). From left to right: evolution times $t=0,\,2^{-14},\,2^{-13},\,2^{-10},\,2^{-8},\,2^{-7}$, respectively.
  • Figure 5: Evolution by Willmore flow for a cube surface and a thick disk surface with spatial resolution $n = 64$, kernel size $n_K =17$, interface parameter $\varepsilon=2^{-5}$, inner stepsize $\tilde{\tau} = 2^{-12}$, and Willmore flow stepsize $\tau = 2^{-18}$. Results are shown at time $0,1$, and time $10$, which reproduce concave surface patches as common effects for Willmore.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 2.1: Variational time discretization of Willmore flow FrRuWi11
  • Remark 2.2: Relation to the MBO scheme and semi-implicit time stepping for Allen-Chan flow
  • Remark 2.3
  • Proposition 2.4
  • proof