Machine Learning VOF Method

• Machine-Learning based VOF is a data-driven method that uses neural networks to predict flux fractions in multiphase fluid flows.
• It replaces traditional geometric reconstructions with fast, direct neural evaluations, offering robust performance for complex 3D interfaces.
• Key strategies include synthetic dataset generation and enforcement of physical constraints to ensure conservation and interface fidelity.

A machine-learning based Volume of Fluid (VOF) method refers to any approach that employs machine learning models—typically neural networks or other data-driven algorithms—to improve or replace components of the classical VOF methodology for simulating multiphase or multi-material fluid flows. Unlike traditional VOF techniques, which rely on algebraic or geometric interface reconstructions and explicit formulas for advection and curvature estimation, ML-based methods learn mappings from synthetic or high-fidelity data, often leading to improved computational efficiency, accuracy, and flexibility in handling complex interface geometries, especially in three dimensions.

1. Methodological Framework

Machine-learning based VOF methods commonly depart from the standard VOF pipeline by replacing explicit interface reconstruction or flux computation steps with neural networks. The core methodological advance is the direct prediction of physically relevant quantities—such as the cell face flux fraction—based only on local discrete information, typically the volume fractions within a cell-centered stencil, and in some cases, local flow parameters (e.g., Courant number). This results in a workflow that bypasses both iterative geometric reconstructions (e.g., PLIC) and the need for analytical expressions for the interface location.

The neural network used is typically a fully-connected feed-forward structure. For three-dimensional domains with a stencil of size $m^3$ and a normalized Courant number $\beta$ for the considered direction, the network accepts a vector of length $m^3 + 1$ and outputs a single flux fraction. The depth (number of hidden layers) and the width (number of neurons per layer) are selected to balance the representation power and computational cost, with ReLU as the most common activation function. Training is performed on synthetic datasets generated through randomized geometric parametrizations of interface shapes, ensuring coverage of both regular and irregular interface configurations (2507.05218).

2. Synthetic Dataset Generation and Training Strategies

Synthetic data generation is essential due to the lack of analytic expressions for flux or curvature in arbitrary interface configurations. For 3D VOF-ML schemes, the training set is created by sampling a large spectrum of interface geometries:

• Half-spaces (planes): The fluid region is a cube intersected by a random plane, with the normal sampled uniformly from the unit sphere.
• Multiple intersecting planes: Yielding complex, possibly non-smooth interfaces.
• Ellipsoidal regions: Incorporating smoother shapes, generated by parameterizing ellipsoid centers and axes, sometimes discretized into polytopes (e.g., with a Fibonacci spiral).
• Latin Hypercube Sampling: Efficiently spans the high-dimensional parameter space.

For each random configuration, the “exact” flux fraction over a face is determined by geometrically integrating the fluid region’s indicator function over the swept face for the prescribed Courant number. The network is trained using mean squared error loss, often first using ADAM and then refining with BFGS optimization (2507.05218). The dataset also includes varying mesh resolutions and interface irregularities, both to enhance generalization and to make the method robust to the kinds of scenarios encountered in realistic multiphase flows.

3. Flux Prediction and Algorithmic Integration

Unlike geometric VOF algorithms (PLIC, MOF, etc.), which reconstruct the interface explicitly via parametrized surfaces and then compute fluxes by geometric integration, the ML-based VOF computes the flux fraction through a face by directly evaluating the trained neural network. The process is as follows:

1. At runtime, for a given cell and face (e.g., $x$-direction), extract the local $m^3$ stencil of adjacent volume fractions.
2. Form an input vector by concatenating these values with the normalized Courant number ($\beta_x = u_x\,\Delta t/\Delta x$).
3. Compute the output flux fraction via a forward pass through the trained network: $\alpha^{(x, \beta)}$.
4. Use this flux in the directional-split finite volume update formula, e.g.,

$$\alpha^{x}_{i,j,k} = \alpha_{i,j,k} - \beta_x (\alpha_{i+1/2,j,k} - \alpha_{i-1/2,j,k}),$$

and similarly for $y$ and $z$ directions.

1. Apply post-processing operators to enforce physical constraints, such as bounding ($\alpha\in[0,1]$), symmetry averaging over cell permutations, and phase complementarity (ensuring additivity when swapping roles of two fluids).

By eliminating iterative and geometric reconstruction steps, the computational pipeline is streamlined; the only runtime cost is the neural network evaluation, which is highly parallelizable and suitable for vectorized hardware acceleration.

4. Enforcement of Physical Constraints

Ensuring that the ML-predicted fluxes uphold conservation and physical symmetry is critical. Several strategies formalized in the method (2507.05218) include:

• Symmetry Averaging: The local stencil exhibits octahedral symmetry due to the cubic geometry of the grid. The flux prediction is averaged over the subgroup of symmetry operations (rotations and reflections) that leave the physics unchanged, guaranteeing model equivariance.
• Phase Additivity (Complementarity): The output is further symmetrized to ensure that swapping fluid phases yields $\alpha + (1 - \alpha^M) = 1$ (where $M$ denotes a complementary stencil).
• Bound Projection: The final predicted flux is projected into admissible interval $[m, M]$ (where $m$ and $M$ are minimum and maximum values ensuring volume fraction stays within $[0, 1]$ after the update).

These steps ensure invariance and mass conservation that typically require geometric proof in analytic VOF schemes.

5. Numerical Performance and Convergence Properties

The ML-based VOF method demonstrates empirical convergence as the mesh is refined, with performance superior to classical upwind and limited downwind schemes. For example, when advecting complex interfaces with prescribed velocity fields, the method yields an $L_1$ error scaling with mesh size as $\approx 1.01$ (VOF-ML), compared to $0.58$ for limited downwind and $0.16$ for upwind schemes. The number of mixed cells (cells with $0 < \alpha < 1$) remains stable, indicating the scheme does not add artificial diffusion or unphysical oscillations during advection.

The neural network’s ability to generalize to irregular interface geometries (not seen during training) is achieved by the inclusion of ellipsoids, planar intersections, and advanced sampling strategies. The method avoids the formation of spurious interface fragments and demonstrates good preservation of the interface shape and volume, crucial for mass conservation in practical CFD simulations.

6. Mathematical and Algorithmic Summary

Neural Network Flux Computation

For an input stencil $x \in \mathbb{R}^{m^3}$ and Courant number $\beta$: $x_0 = [x;\, \beta]$

$$x_l = \rho(A_l x_{l-1} + b_l) \quad (l = 1, \dots, L-1)$$

$\alpha^{(x, \beta)} = A_L x_{L-1} + b_L$

where $\rho(z) = \max(0, z)$ is the ReLU.

Physical Constraints

• Symmetry averaging:

$$\hat{\alpha}^{(x, \beta)} = \frac{1}{8} \sum_{j=1}^8 \alpha^{(\sigma_j x, \beta)}$$

with $\sigma_j$ symmetry operations.

• Phase additivity:

$$\bar{\alpha}^{(x, \beta)} = \frac{1}{2} \alpha^{(x, \beta)} + \frac{1}{2} [1 - \alpha^{(M x, \beta)}]$$

• Projection:

$$\text{Final flux: } \alpha_{i+1/2,j,k}^{(\text{final})} = P_{[m_{i,j,k}^x, M_{i,j,k}^x]}(\bar{\alpha}^{(x, \beta)})$$

where $P$ projects onto $[m, M]$ computed from CFL and local volume-fraction constraints.

7. Context, Applications, and Future Directions

The machine-learning based VOF framework establishes a new paradigm for multiphase CFD by enabling direct flux computation from local field data, obviating the need for explicit interface reconstructions that dominate classical methods. Its main advantages include improved convergence, robustness for irregular interface geometries, and compatibility with hardware acceleration.

This method is especially promising for:

• Large-scale 3D multiphase flows with complex interface dynamics.
• Scenarios where traditional geometric operations are expensive or challenging (e.g., multi-material flows, topological changes).
• Integration with neural-accelerated simulation stacks, where the neural network forward pass is efficiently parallelized.

Ongoing and future research directions include extending the dataset for greater generality (e.g., more singular interfaces), integrating uncertainty quantification around ML-based predictions, and formalizing adaptive schemes that locally select between ML and analytical flux computations for optimal performance and robustness.

This comprehensive approach elevates the ML-based VOF scheme as a rigorous, accurate, and efficient alternative to classical interface-capturing techniques for multi-material and multiphase CFD (2507.05218).