Learning the Willmore flow with a lightweight neural operator

1 Introduction

• Fourth-order geometric flow is challenging even for smooth interfaces, and numerical methods must resolve curvature variations while accommodating topology changes.Thin-interface resolution and fourth-order dynamics also make phase-field computation demanding.Source: §1 · p. 2, §1.1 · p. 2

1.1 Phase-field formulation

• Phase-field methods encode interfaces as smooth transitions, with Allen–Cahn flow having motion by mean curvature as its sharp-interface limit.For Willmore flow, the phase profile is described using signed distance; a non-oriented encoding instead uses an even profile of distance to the interface.Source: §1.1 · pp. 2–3

1.2 Related numerical and learning approaches

• Existing learning approaches range from physics-informed methods to general operator learning and equation-specific architectures, while RSS still requires optimization at each time step.The equation-specific family approximates a solution operator using structure from a discretization scheme.Source: §1.2 · pp. 3–4

1.3 Contributions and outline

• The paper extends a lightweight, single-step neural operator trained on RSS trajectories to oriented and non-oriented Willmore flow in two and three dimensions.Once trained, the network replaces the optimization at each RSS time step with one evaluation.Source: §1.3 · p. 4
• The contributions include a compact architecture motivated by an Allen–Cahn semigroup expansion, radial kernels that remedy anisotropic non-oriented dynamics, and transfer of trained 2D radial Fourier profiles to initialize 3D models.The transferred model serves as a regularizer for reconstruction from unoriented point clouds, without additional 3D training.Source: §1.3 · p. 4

2.1 Mean curvature network building blocks

• Fixed local stencils let the networks act on arrays of different sizes, provided the ratio ε/h is preserved.This compatibility condition is used for reconstruction on another grid but does not itself establish convergence under mesh refinement.Source: §2.1 · p. 6

2.2 Architecture motivated by the Allen–Cahn semigroup

• Parallel Allen–Cahn branches form a Willmore operator whose diffusion scales and recombination weights are optimized during training.The construction cancels the Cahn–Hilliard contribution while retaining Willmore gradient descent; stability still depends on suitable parameter choices.Source: §2.2 · pp. 6–8

2.3 Willmore flow approximation using the hybrid RSS approach

• The RSS scheme deforms a π-shaped curve toward circular shapes while preserving crossing singularities, and remains well-behaved at eτ=τ.The scheme minimizes an energy combining a Willmore minimizing-movement term with a mean-curvature-network surrogate, using ADAM for its inner iterations.Source: §2.3 · p. 9
• At eτ=τ/64, J=1000 RSS iterations leave about 15% mean relative error, while roughly 2000 iterations are needed for satisfactory accuracy.Increasing J beyond 2000 brings no further gain because time-discretization error dominates; reducing the error further requires a smaller eτ.Source: §2.3 · p. 10, Table 1 · p. 11

2.4 Dataset generation and training procedure

• The 2D database covers disks, ovals, unions of two or three disks, and star-shaped curves, yielding 5346 consecutive-triplet training examples.Each sample is a 101-state subsequence from a 100-step evolution; the 80/20 train-validation split is by evolution to avoid trajectory leakage.Source: §2.4 · pp. 11–12
• Because Willmore evolution is slow, training used a multistep loss with S = 2, also adopted for non-oriented curve and surface applications in 3D.Source: §2.4 · p. 13
• The oriented-curve models’ validation losses were 4.73 × 10−7 and 3.71 × 10−7, a 21% gap too small by itself to justify preferring either architecture.The losses should be compared within a table, not across tables, because datasets, time steps, and interface representations differ.Source: §2.4 · p. 13, Table 3 · p. 14

2.5 Application for the approximation of the Willmore flow

• The network generalized to smooth interfaces and unseen complex configurations, with evolutions visually close to RSS at significantly lower computational cost.RSS solves an optimization problem at each step, requiring J evaluations of SNN; each network step requires one evaluation of W NN.Source: Figure 10 · p. 14, Figure 12 · p. 15, §2.5 · p. 15
• The reported speed-up was not an artifact of choosing J = 300: reducing J to 30 degraded the dynamics, and accurate RSS evolution for a circle may require J around 2000.For the smooth flower, J = 30 produced markedly slower evolution that was no longer the Willmore flow; the fully converged reference would therefore be several times more expensive.Source: §2.5 · p. 16

3.1 Dataset, training and fast approximation of the Willmore flow

• Applying Ψ[u] = −ε|∇u| to oriented RSS trajectories produces non-oriented training data concentrated near the interface and decaying on both sides.The profile uses unsigned distance, so it does not distinguish inside from outside.Source: §3 · p. 16, §3.1 · p. 17
• The 2-parallel model's losses remain near 1.8 × 10−3, while the 4-parallel model decreases by four orders of magnitude.With four time parameters, the next-order error can be cancelled; only the 4-parallel model is retained for later tests.Source: §3.1 · p. 17
• The free-kernel model develops axis-favored contraction that amplifies into spurious oscillating interfaces, despite low validation loss.Five of its ten learned kernels are visibly anisotropic, with a preference for grid axes.Source: §3.1 · p. 18, Figure 15 · p. 18, Figure 16 · p. 19

3.2 Enforcing isotropy: rotationally invariant diffusion kernels

• The isotropic architecture learns a radial profile with a shared MLP, assigning equal coefficients to kernel positions at the same normalized radius.The profile is represented by a three-layer MLP with 57 learnable parameters.Source: §3.2 · pp. 19–20, Figure 17 · p. 20
• Radial kernels yield stable curve evolution toward a circle without spurious structures, though a residual hexagonal grid effect remains.The remaining directional bias is attributed to sampling the radial kernel on a Cartesian stencil and repeated application of the operator.Source: §3.2 · p. 22, Figure 21 · p. 23
• The isotropic model produces a triple junction after training only on domain boundaries, outside the configurations represented in its training data.The non-oriented representation does not distinguish inside from outside.Source: §3.2 · p. 23

3.3 Application to reconstruction of curves from point clouds

• For point-cloud reconstruction, the learned Willmore time-step map supplies curvature regularization in a forward pass, while a pointwise minimum attaches the evolving field to the data.The parameter α controls the constraint: α = 1 imposes it fully, while α > 1 allows distance from noisy samples.Source: §3.3 · p. 24
• With noisy circle samples, smaller α keeps reconstructed curves closer to points; a double-bubble reconstruction develops a triple junction.The double-bubble model was trained only on oriented configurations.Source: §3.3 · p. 25, Figure 23 · p. 25, Figure 24 · p. 26

4.1 Construction of a dataset in 3D

• The training database uses RSS trajectories from cubes, cuboids, ellipsoids, unions of two spheres, and tori on a 64^3 grid.These shapes provide strongly varying mean curvature, non-trivial topology, and a singularity in the union-of-two-spheres case; the stationary sphere is excluded because its stability can be learned from near-spherical examples.Source: §4.1 · pp. 26–27

4.2 Training and application to the approximation of oriented Willmore flow

• The 3D rotationally invariant diffusion kernels are defined by radial profiles, so the network trains radial MLPs rather than full convolution kernels.A learnable output scale and a loss on first-order differences help fit the update magnitude; omitting the second loss term left the phase field virtually unchanged.Source: §4.2 · pp. 27–29
• The learned network closely matches the 3D RSS flow while reducing computation time significantly.The comparison uses a smooth flower initial surface; both schemes reduce Willmore energy and appear very similar.Source: §4.2 · pp. 29–30, Table 8 · p. 30, Figure 30 · p. 30

5.1 Transferring the trained 2D kernels to three dimensions

• Non-oriented 3D networks could not be trained from scratch in practice, so the method initializes them from trained 2D kernels and fine-tunes on 3D data.The transfer preserves learned radial Fourier profiles, copies pointwise parameters, and maps transferred kernels back to a radial parameterization.Source: §5 · p. 31, §5.1 · p. 32
• A 2D-to-3D transfer alone maintains a coherent interface without additional 3D training, but its evolution does not exactly match the reference.The transferred spiky-ball interface appears to converge toward a sphere, while the trajectory comparison suggests retraining is needed.Source: §5.1 · p. 32, Figure 32 · p. 32

5.2 Training and results

• The fine-tuned rotationally invariant model gives a good approximation of the Willmore flow in the tested unoriented 3D example.The transfer-based initialization reduced training to a few epochs; the example's evolution is similar to the oriented model's flow.Source: §5.2 · p. 33, Figure 33 · p. 33

5.3 Application to surface reconstruction from point clouds

• For point-cloud reconstruction, the transferred model acts as a curvature regulator without requiring an additional 3D training database.The procedure uses iterations of (3.3) followed by unconstrained network applications; the regularization subproblem is handled by a single network evaluation.Source: §5.3 · p. 33
• The reconstruction procedure extends to unoriented point clouds and can produce smooth surfaces that preserve a torus's hole and global topology.The paper also demonstrates a double bubble with a triple junction without prescribed junction normals, and reports observed hole-closing on a partial sphere point cloud without establishing a general guarantee.Source: §5.3 · pp. 34, 36, Figure 34 · p. 34, Figure 36 · p. 35

• The authors developed the scientific ideas, methods, analyses, and interpretations.Source: Declaration of generative AI use · p. 36
• AI tools assisted with language editing and manuscript presentation.Source: Declaration of generative AI use · p. 36
• AI tools also assisted with generating and revising parts of the computational code.Source: Declaration of generative AI use · p. 37

• The implementation extends nnpf model classes and adds custom data-generation procedures for the experiments.Source: Code and data availability · p. 37
• The nnpf package is built on PyTorch Lightning.Source: Code and data availability · p. 37
• The nnpf package is available on GitHub.Source: Code and data availability · p. 37