⚡ electrical engineering

Spectral-Embedded Operator Learning for Three-Phase Interfacial Flow: A Ternary Cahn-Hilliard-Navier-Stokes Benchmark

This paper introduces a three-phase interfacial flow benchmark based on a ternary Cahn-Hilliard-Navier-Stokes system and demonstrates that a spectral-embedded operator learning architecture using a fixed Chebyshev dictionary significantly outperforms standard and Fourier-based DeepONet variants in predicting complex multiphase dynamics.

Original authors: Muhammad Abid, Arth Sojitra, Omer San

Published 2026-09-24

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CC BY 4.0

Original authors: Muhammad Abid, Arth Sojitra, Omer San

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the complex world of fluid dynamics, where liquids and gases mix and separate, some of the most difficult challenges involve three distinct substances interacting at once. Imagine a bubble of air rising through water, only to punch through a layer of oil floating on top. This scenario is not just a visual curiosity; it is a fundamental problem in industries ranging from oil recovery to carbon storage. The difficulty lies in the fact that these three fluids do not just touch; they meet at a single, moving point where their surfaces balance against one another. When the bubble breaks through the oil layer, it drags a thin column of water upward, creating a complex, shifting shape that is incredibly hard to predict with standard computer simulations. These simulations are so computationally expensive that running them thousands of times to test different conditions is often impossible, leaving engineers without the data they need to design better systems.

To solve this, researchers have turned to a new kind of artificial intelligence called operator learning. Instead of teaching a computer to solve a single physics problem, they teach it to learn the entire rulebook that governs how the fluids behave across thousands of different scenarios. However, a major question remained: does the way the computer "sees" the physical space matter when the problem involves three fluids and complex, moving boundaries? A team of researchers at the University of Tennessee set out to answer this by creating a rigorous test case. They simulated a thousand different versions of an air bubble rising through water and piercing an oil layer, generating a massive library of reference data. They then trained three different versions of an AI model on this data. The only difference between the three models was how they processed the coordinates of space and time. One model looked at the raw numbers, another used a method based on repeating waves, and the third used a method based on smooth, non-repeating curves.

The results revealed that the choice of how the computer views space is critical. The model that used the smooth, non-repeating curves, known as a Chebyshev dictionary, significantly outperformed the others. It reduced the error in its predictions by nearly a quarter compared to the model using raw numbers and by about seventeen percent compared to the wave-based model. This improvement was not just a small tweak; it happened across every single output the model was asked to predict, including the speed of the flow and the exact position of the fluid boundaries. The researchers found that the wave-based model, which is often excellent for problems that repeat themselves like sound or light, struggled here because the bubble's rise and the oil's deformation are one-way, non-repeating events. The smooth curve method, by contrast, is naturally suited to describing things that move steadily in one direction without looping back.

The study also showed that the superior model was particularly good at capturing the most difficult moments of the simulation: the exact instant the bubble breaks through the oil and the thin, fragile column of water it drags along. In these critical moments, the other models made larger mistakes, blurring the sharp edges of the fluids. The winning model kept these edges crisp and accurate. Furthermore, the researchers ensured that their AI respected a fundamental rule of physics: that the three fluids must always add up to a total volume of one. They built this rule directly into the model's structure, so the AI could never violate it, ensuring that its predictions were physically possible. This approach allowed them to prove that any small errors the AI made were due to the model itself, not a failure of the underlying physics.

Ultimately, this work demonstrates that for complex, multi-fluid problems, the architecture of the AI matters as much as the data it learns from. By choosing a coordinate system that matches the physical reality of the flow—specifically one that handles steady, non-repeating motion well—researchers can build surrogates that are far more accurate and reliable. This finding suggests that for future engineering applications, simply throwing more data at a standard model is not enough; the model must be designed to understand the specific nature of the movement it is trying to predict. The study provides a new benchmark for the field, proving that with the right mathematical tools, artificial intelligence can master the intricate dance of three-phase flows, offering a faster and more precise way to simulate the complex interactions that drive modern industry.

Technical Summary: Spectral-Embedded Operator Learning for Three-Phase Interfacial Flow

Problem Statement
This paper addresses a gap in Scientific Machine Learning (SciML) benchmarks: the lack of operator-learning evaluations for constrained, multiphase flows involving three immiscible phases. While existing benchmarks focus on single-field or two-phase problems, three-phase flows introduce qualitative difficulties, including three pairwise interfaces meeting at a triple junction governed by Neumann-triangle balance, spontaneous phase film formation, and algebraic constraints where phase variables must sum to unity pointwise. The specific physical problem modeled is an air bubble rising through water, piercing a water-oil interface, and entraining a water plume into the oil within a bounded, wall-confined domain. This configuration encapsulates thin films, high curvature, moving triple junctions, and topological transitions.

Methodology
The authors introduce a benchmark based on a Ternary Cahn-Hilliard-Navier-Stokes (CHNS) system.

Reference Solver: A structure-preserving solver generates 1,024 reference simulations sampled via Sobol sequences across a nine-dimensional parameter space (including Eötvös number, Reynolds number, spreading coefficients, density/viscosity ratios, and initial bubble position). The solver algebraically preserves the Gibbs simplex constraint ( $\sum c_i = 1$ ) by choosing mobilities inversely proportional to spreading coefficients, making the constraint an identity of the discretization rather than an approximation.
Operator Learning Framework: The study employs the Deep Operator Network (DeepONet) architecture to learn the mapping from the nine physical parameters to a five-channel space-time field (water, oil, and air volume fractions, and two velocity components).
Controlled Comparison: Three parameter-matched DeepONet variants are compared. They share identical branch networks, trunk depths, widths, latent dimensions, optimizers, and training data. The only variable is the trunk coordinate embedding ( $\phi$ $ϕ$ ):
1. DeepONet (Baseline): Uses raw, affinely rescaled coordinates $(x, y, t)$ .
2. FEDONet: Uses Random Fourier Features (RFF) to embed coordinates into a sinusoidal dictionary.
3. SEDONet: Uses a fixed, deterministic tensor-product Chebyshev dictionary.
Structural Constraints: To respect the physical constraint $\sum c_i = 1$ , the model predicts only two concentration channels ( $c_w, c_o$ ) and structurally closes the third ( $c_a = 1 - c_w - c_o$ ) using an affine map that accounts for channel standardization.

Key Contributions

A New Benchmark: The first operator-learning benchmark for three-phase flow with a simplex-constrained state and a moving triple junction, featuring a five-channel output over a nine-dimensional parameter space.
Structure-Preserving Reference Data: A reference solver that enforces the simplex constraint to machine precision via algebraic design, allowing any violation in the surrogate to be unambiguously attributed to the learning model.
Graded Truncation: A method for ordering the Chebyshev dictionary in three dimensions (total-degree ordering) to avoid the defects of lexicographic cropping, ensuring a degree-balanced dictionary.
Parameter-Exact Comparison: A rigorous comparison where FEDONet and SEDONet have identical parameter counts, isolating the effect of the coordinate representation.
Error Localization: A decomposition of errors spatially, temporally, and by channel to identify where architectural changes yield benefits, moving beyond scalar error norms.

Results

Accuracy: SEDONet (Chebyshev) reduces the mean relative $L_2$ error by 24.0% compared to the raw-coordinate DeepONet and by 16.8% compared to FEDONet. This improvement is uniform across all five output channels.
Mechanism of Improvement:
- FEDONet primarily improves channels with oscillatory or small-scale structures (air volume fraction $c_a$ and velocities), consistent with its periodic prior.
- SEDONet improves all channels, including the large-region concentration fields ( $c_w, c_o$ ). The paper attributes this to the Chebyshev dictionary's ability to represent monotone, non-periodic trends (e.g., bubble rise, plume advancement) efficiently with low-order modes, whereas periodic dictionaries require slowly converging series for such trends.
Spatial/Temporal Distribution: Error gains are localized to the diffuse interfaces and occur after the bubble breakthrough (when the topological transition and triple junction appear). The error is negligible at the walls, falsifying the hypothesis that Chebyshev gains are due to boundary-layer resolution in this specific problem.
Constraint Preservation: All models satisfy the simplex constraint to the single-precision rounding floor due to the affine closure, confirming that the constraint is handled structurally rather than learned.

Significance and Claims
The paper claims that for strongly non-periodic, wall-confined, interface-dominated flows, a Chebyshev spectral embedding in the trunk is superior to both raw coordinates and random Fourier features. The significance lies not merely in the error reduction but in the identification of the mechanism: the polynomial dictionary effectively captures the global, monotone drift of the flow while maintaining the capacity to resolve sharp interfaces.

The authors are modest regarding the scope of these claims:

The results are specific to the diffuse-interface regime at a finite Cahn number; transfer to sharp-interface physics is assumed only insofar as the diffuse solution approximates it.
The design space is restricted to a single topological class; extrapolation across regime boundaries (e.g., total wetting) is untested.
The study relies on single training runs per model; the authors acknowledge that repeated runs are necessary to establish statistical significance.
The improvement is attributed to the coordinate representation's ability to condition the optimization landscape and represent specific flow trends, not to an increase in model capacity (as FEDONet and SEDONet are parameter-identical).

The work concludes that while coordinate embeddings are a minimally invasive intervention, their efficacy is contingent on the problem's geometry and temporal structure, and they do not overcome the fundamental Kolmogorov width limitations of linear reduced-order modeling for advection-dominated problems.

Technical Summary: Spectral-Embedded Operator Learning for Three-Phase Interfacial Flow

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