Neural ODE Flows for Mesh Deformation

• Neural ODE Flows are diffeomorphic transformations defined by the continuous integration of a neural velocity field, ensuring smooth, topology-preserving mesh deformations.
• They model vertex-wise trajectories using probabilistic mesh encodings and contextual features, enabling flexible applications in CAD, geometry processing, and biomedical analysis.
• Optimized with loss functions like the sliced Wasserstein distance, Neural ODE Flows yield robust quantitative and qualitative improvements over classical deformation methods.

Neural Ordinary Differential Equation (ODE) Flows represent a class of deformation models for surfaces, especially meshes, that use neural parameterizations of continuous-time diffeomorphic transformations. These models treat the mesh as a set of points flowing along trajectories defined by a learned, time-dependent velocity field, ensuring topology-preserving, smooth, and bijective deformations. Neural ODE Flows have found significant traction in geometry processing, biomedical shape analysis, and computer-aided design, providing both theoretical guarantees and tangible improvements over classical methods in terms of flexibility, regularity, and robustness.

1. Definition and Mathematical Formulation

Neural ODE Flows define a deformation as the continuous integration of a neural velocity field over time. For a mesh with vertices $x(0) \in \mathbb{R}^3$, the evolution is governed by

$$\frac{dx(t)}{dt} = v_\phi(x(t), t), \quad x(0) = x_0,$$

where $v_\phi: \mathbb{R}^3 \times [0,1] \to \mathbb{R}^3$ is a multi-layer perceptron (MLP) with parameters $\phi$ (Le et al., 2023, Huang et al., 2020). The flow map at $t=1$, $x(1) = \mathcal{D}(x_0)$, is a diffeomorphism under mild regularity assumptions. Consequently, the entire mesh undergoes a globally invertible mapping, preventing self-intersections and guaranteeing smoothness of the deformation.

2. Representation of Meshes and Trajectory Modeling

Neural ODE-based deformations require a robust scheme for representing and comparing meshes:

• Probability Measure Encodings: Surfaces are encoded as probability measures, such as area-weighted triangle distributions, empirical measures via Monte Carlo sampling, or as discrete varifolds (Dirac measures at triangle barycenters, possibly with normals) (Le et al., 2023). These encodings admit continuous and discrete optimal transport-based discrepancies.
• Vertex-wise Trajectories: Each mesh vertex evolves independently along the neural ODE’s trajectory, maintaining the original mesh’s connectivity (face structure), which is crucial for applications in computer-aided design (CAD) and medical imaging (Huang et al., 2020).
• Feature Integration: The velocity field can consume geometric and contextual features—such as MRI patches in biomedical settings—alongside the current spatial position and time, allowing for data-adaptive and context-aware deformations (Le et al., 2023).

3. Loss Functions and Training Objectives

Optimization of Neural ODE Flows typically utilizes discrepancy measures that are robust to nonuniform sampling and misalignment:

• Sliced Wasserstein Distance (SWD): The primary metric is the Monte Carlo-approximated SWD between varifold measures of the predicted and target surfaces. SWD projects measures onto random 1D directions, sorts and matches samples, and averages the 1D optimal transport costs: $$SW_p^p(\mu, \nu) = \mathbb{E}_{\theta \sim \text{Unif}(S^{2})}\left[ W_p^p(\theta_\#\mu, \theta_\#\nu) \right].$$ This yields $O(m \log m)$ complexity, circumventing the $O(m^2)$ cost of Chamfer or full Wasserstein metrics (Le et al., 2023).
• Iterative Closest Point (ICP) and Rigidity Terms: In some variants, fitting (symmetrized Chamfer) and rigidity (as-rigid-as-possible, ARAP) energies are incorporated to ensure both alignment and plausible local behavior (Huang et al., 2020).

No explicit supervision for intermediate trajectories $x(t)$ is necessary; only the pairwise registration loss at $t=1$ is minimized, leading to continuous and meaningful interpolations between shapes.

4. Properties and Theoretical Guarantees

Neural ODE Flows for deformation inherit several critical properties:

• Diffeomorphic Guarantee: Provided the neural velocity field is Lipschitz-continuous, the flow is guaranteed to be a diffeomorphism, ensuring global injectivity and preventing mesh self-intersections (Le et al., 2023, Huang et al., 2020).
• Invertibility and Smooth Interpolation: The inverse deformation is readily computable by integrating the reversed velocity field backward in time. Intermediate shapes at $t \in [0,1]$ are valid, plausible deformations between the source and target.
• Scalability: By using advanced loss formulations (varifold/SWD) and efficient mesh encoding, the approach scales linearly with mesh complexity in both computational and memory resources (Le et al., 2023).

5. Algorithmic Pipeline and Implementation

A standard pipeline for Neural ODE Flow-based mesh deformation contains:

1. Preprocessing: Input meshes are normalized and preprocessed, possibly with feature-aware subdivision and skeleton coarsening to produce uniform, high-quality connectivity and robust local graph structures (Huang et al., 2020).
2. Velocity Field Learning: The velocity field $v_\phi$ is modeled by an MLP, often receiving spatial and contextual features.
3. ODE Integration: Numerical integration (e.g., Dormand-Prince RK45) is performed on all mesh vertices to obtain the deformed geometry. Gradients are propagated through this integration via adjoint sensitivity methods (Le et al., 2023).
4. Loss Computation: Meshes are compared in probability measure space using SWD or similar, backpropagating losses through the ODE solver.
5. Postprocessing: The deformation can be transferred to fine meshes through a template or skeleton mapping, preserving geometric detail.

6. Comparative Analysis and Empirical Results

• Quantitative Metrics: Neural ODE Flow methods achieve lower Earth Mover’s Distance (EMD), lower SWD, lower average symmetric surface distance (ASSD), higher Chamfer Normals, and near-zero self-intersection rates compared to prior diffeomorphic, ICP, or deep learning baselines (Le et al., 2023).
• Qualitative Performance: Deformations are free of artifacts (crowding, fold-overs) even under large shape changes; the approach enables smooth animations and design interpolations not constrained by template correspondence or rigid cages (Huang et al., 2020).
• Downstream Utility: Improved fitting in scan-to-CAD registration, texture mapping, and synthetic data generation; robust even in the presence of mesh topological noise due to carefully designed pipelines (Huang et al., 2020).

7. Limitations and Future Directions

While Neural ODE Flows provide theoretically robust and empirically superior mesh deformation capabilities, several limitations persist:

• Solver and Pipeline Complexity: Numerical ODE integration and adjoint backpropagation introduce additional computational barriers compared to static methods, though the overall time scales favorably with mesh size (Le et al., 2023).
• Expressivity vs. Regularization: Balancing flexibility (capturing large, nonlinear deformations) and topology preservation sometimes requires careful velocity network tuning and regularization.
• Data Modalities: Successes are prominent in domains with reliable geometric or volumetric context features; generalization to highly anisotropic, topologically changing, or richly textured surfaces is an area of active research.

Neural ODE Flows form a foundational paradigm for diffeomorphic and topology-preserving mesh deformation, synthesizing elements of neural network design, geometric measure theory, and numerical analysis, and replacing ad hoc, non-bijective transformations with rigorously regularized, data-driven flows (Le et al., 2023, Huang et al., 2020).