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

Muhammad Abid; Arth Sojitra; Omer San

physics.flu-dynAug 29, 2026

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

Authors: Muhammad Abid, Arth Sojitra, Omer San

Organizations: Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville Knoxville, TN 37996, USA.

Abstract

Operator-learning surrogates have been benchmarked largely on single-field, single-interface problems, leaving unclear whether architectural choices validated in those settings transfer to constrained, multiphase flows. We introduce a three-phase interfacial-flow benchmark to examine whether the trunk coordinate representation matters for a multi-channel, interface-dominated target. The configuration consists of 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. Reference data are generated using a structure-preserving ternary Cahn-Hilliard-Navier-Stokes solver that algebraically preserves the simplex constraint. From 1,024 Sobol-sampled simulations spanning a nine-dimensional parameter space, we learn the mapping from physical parameters to five-channel space-time fields. We compare three parameter-matched DeepONet variants differing only in trunk representation: raw coordinates (DeepONet), random Fourier features (FEDONet), and a fixed tensor-product Chebyshev dictionary (SEDONet). SEDONet reduces the test relative L2 error by 16.8% compared with FEDONet and by 24.0% compared with DeepONet, while improving all five output channels. Spatial and temporal error analyses localize the principal gains near the diffuse interfaces and after bubble breakthrough. The results indicate that the Chebyshev representation is particularly effective for the strongly non-periodic wall-normal and temporal structure of this three-phase flow.

Figures & tables

Figure 1: The target physics: RGB composite of the three volume fractions (red $=$ water, green $=$ oil, blue $=$ air) at $t=0,\,0.75,\,1.50,\,2.38,\,3.12,\,4.00$ . The bubble rises, deforms into a spherical cap, pierces the water-oil interface, and entrains a water plume that thins and pinches as the bubble detaches.

	DeepONet	FEDONet	SEDONet
Embedding $\phi$	$\boldsymbol{\xi}$	$[\sin,\cos](2\pi B\boldsymbol{\xi})$	$\{T_{k}T_{l}T_{m}\}_{\mathcal{A}}$
Implied prior	none (spectral bias)	periodic, translation invariant	non-periodic, boundary emphasized
Represents a linear trend	via nonlinearities	slowly converging series	exactly, in two modes

Table 1: The three trunk embeddings side by side. Everything not listed (branch, trunk depth and width, latent width

p

, loss, optimizer, schedule and data split) is identical across the three models.

Figure 2: The three architectures differ only in the trunk coordinate embedding. (a) The branch-trunk factorization of Eq. 16 : the branch encodes the input, the trunk encodes the query coordinate after a fixed embedding $\phi$ , and the two are combined by an inner product. (b)-(d) The three fillings of the $\phi$ slot, raw rescaled coordinates, random Fourier features, and a deterministic Chebyshev dictionary, which constitute the entire experimental manipulation. The first trunk layer has the same width in both (c) and (d), so FEDONet and SEDONet are parameter-identical, and no block carries trainable parameters.

Model	Trunk embedding $\phi$	$d_{\mathrm{trunk}}$	Parameters
DeepONet	raw $\boldsymbol{\xi}=(\xi_{x},\xi_{y},\xi_{t})$	$3$	$0.73$ M
FEDONet	$[\sin(2\pi B\boldsymbol{\xi}),\cos(2\pi B\boldsymbol{\xi})]$ , $M=128$	$256$	$0.86$ M
SEDONet	$T_{k}(\xi_{x})T_{l}(\xi_{y})T_{m}(\xi_{t})$ , graded, $K=10$	$256$	$0.86$ M

Table 2: Model configurations. The three architectures share branch and trunk depths and widths and differ only in the trunk coordinate embedding; the parameter gap is exactly the first trunk layer.

Model	mean	$c_{w}$	$c_{o}$	$c_{a}$	$u$	$v$	$\mathcal{E}_{\Delta}$
DeepONet	0.0821	0.0319	0.0170	0.1331	0.1406	0.0879	$1.79\times 10^{-7}$
FEDONet	0.0750	0.0315	0.0170	0.1236	0.1231	0.0797	$1.79\times 10^{-7}$
SEDONet	0.0624	0.0251	0.0136	0.1053	0.1037	0.0644	$1.79\times 10^{-7}$
SEDONet vs. DeepONet: $-24.0\%$ SEDONet vs. FEDONet: $-16.8\%$ channels improved: $5/5$

Table 3: Test relative

L^{2}

error over

103

held-out simulations, and the maximum simplex violation

\mathcal{E}_{\Delta}

of Eq. 35 . Lowest per column in bold.

Comparison	mean	$c_{w}$	$c_{o}$	$c_{a}$	$u$	$v$
FEDONet vs. DeepONet	$8.6$	$1.3$	$0.0$	$7.1$	$12.4$	$9.3$
SEDONet vs. DeepONet	$24.0$	$21.3$	$20.0$	$20.9$	$26.2$	$26.7$
SEDONet vs. FEDONet	$16.8$	$20.3$	$20.0$	$14.8$	$15.8$	$19.2$

Table 4: Per-channel error reduction (%) implied by Table 3 . The two embeddings differ in where their gains fall as well as in magnitude: the Fourier trunk concentrates its improvement on the oscillatory channels, while the Chebyshev trunk improves every channel by a comparable fifth to a quarter.

Panel of Figure 3 — Figure 3: Test relative $L^{2}$ over the $103$ held-out simulations. (a) Channel-averaged error. (b) Per-channel breakdown. The ordering SEDONet $<$ FEDONet $<$ DeepONet holds for the mean and for every one of the five channels, but the margins are strongly channel dependent: FEDONet is close to the baseline on $c_{w}$ and $c_{o}$ , whereas SEDONet improves those two channels by about a fifth.

Figure 4: Test relative $L^{2}$ of each held-out member (points) against the three most influential design parameters, with binned means (lines), log scale. The SEDONet curve lies below the other two across the entire range of all three parameters and the vertical separation is roughly constant in the log, so the improvement is multiplicative and essentially uniform in $\boldsymbol{\pi}$ rather than concentrated in a subregion of the design.

Figure 5: Top: wall-normal error profile

e(y)

from Eq. 36 at

t=0,1,2,3,4

; the dashed line marks the undisturbed water–oil interface

y_{\mathrm{int}}=0.9

. Bottom: domain-averaged phase error against time, with the vertical dashed line marking the snapshot shown above it. Error is concentrated at the interfaces, near zero in both bulk layers and at both walls, and the three models separate only after breakthrough.

Figure 6: Test member $269$ : RGB composite of $(c_{w},c_{o},c_{a})$ (red $=$ water, green $=$ oil, blue $=$ air) at $t=0,1,2,3,4$ . Rows: reference, DeepONet, FEDONet, SEDONet. All three surrogates reproduce the full sequence, rise, cap deformation, breakthrough, plume entrainment, detachment, and the correct topology at every time, including the reconnected oil layer and the sharp water spike beneath the detached bubble at $t=4$ .

Figure 7: Absolute phase error for the same member on a shared colour scale. The error is a thin shell hugging the bubble rim, the water-oil interface and the plume flanks, and is black over the bulk of both liquid layers. The baseline’s rim is a thick bright ring; SEDONet’s is a thinner, dimmer arc. The one persistent bright feature common to all three is the horizontal band immediately beneath the bubble crown at $t\geq 2$ , where the drainage film between the bubble and the oil is thinnest.

$t$	Model	mean	$c_{w}$	$c_{o}$	$c_{a}$	$u$	$v$
$0$	DeepONet	$\dagger$	0.0337	0.0073	0.1439	$\dagger$	$\dagger$
	FEDONet	$\dagger$	0.0320	0.0093	0.1275	$\dagger$	$\dagger$
	SEDONet	$\dagger$	0.0236	0.0045	0.1056	$\dagger$	$\dagger$
$1$	DeepONet	0.0612	0.0244	0.0067	0.1004	0.1067	0.0679
	FEDONet	0.0618	0.0231	0.0089	0.0865	0.1148	0.0755
	SEDONet	0.0609	0.0249	0.0046	0.1069	0.1075	0.0605

Table 5: Relative

L^{2}

error on test member

269

, resolved by snapshot. The

t=0

velocity entries are undefined, for the reason given in the text, and are marked

\dagger

. Lowest per row in bold.

Model	mean	$c_{w}$	$c_{o}$	$c_{a}$	$u$	$v$
DeepONet	0.1421	0.0546	0.0359	0.2527	0.2233	0.1440
FEDONet	0.1083	0.0427	0.0259	0.1645	0.1892	0.1192
SEDONet	0.0954	0.0350	0.0230	0.1555	0.1624	0.1011

Table 6: Relative

L^{2}

error on member

269

aggregated over the full space–time grid, that is, with a single global denominator rather than a per-snapshot one. Errors are roughly

50\%

above the test mean of Table 3 and the overall ordering is restored.

Figure 8: Absolute deviation of two derived diagnostics computed from the predicted fields and compared with the reference: bubble centroid height $y_{\mathrm{bubble}}$ (left) and entrained plume volume $V_{\mathrm{plume}}$ (right). Lines are means over the $103$ test members, bands the interquartile spread.

Model	Test rel. $L^{2}$	Reduction	Parameters
DeepONet (raw trunk)	0.0821	—	0.73 M
FEDONet (Fourier)	0.0750	$8.6\%$	0.86 M
SEDONet (Chebyshev)	0.0624	$24.0\%$	0.86 M

Table 7: Summary. All three models share branch and trunk depths, widths, optimizer, schedule and data, and differ only in the trunk embedding

\phi

. The FEDONet–SEDONet comparison is parameter-exact.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

$\sigma_{B}$	$0.05$	$0.25$	$0.5$	$1.0$	$2.0$	$4.0$	$8.0$
$\kappa(G)$	$2\times 10^{20}$	$5\times 10^{15}$	$5\times 10^{8}$	$3.1\times 10^{2}$	$3.4$	$1.5$	$2.3$

Appendix

Table 8: Conditioning of the Fourier feature Gram matrix on the query grid (

M=128

). The graded Chebyshev dictionary on the same grid gives

\kappa(G)\approx 2.7\times 10^{2}

with no parameter to select.

Domain $\Omega$	$(0,1)\times(0,2.5)$
Simulation grid	$128\times 320$ ( $h=1/128$ )
Output grid (2 $\times$ block mean)	$64\times 160$
Bubble diameter $d$	$0.25$
Interface height $y_{\mathrm{int}}$	$0.9$
Gravity $g$ , water density $\rho_{w}$	$0.98$ , $1$
Velocity scale $U=\sqrt{gd}$	$0.4950$

Appendix

Table 9: Discretization, fixed physical quantities and cost. With Table 10 this specifies the benchmark completely.

#	Symbol	Meaning	Range	Sets
1	$\mathrm{Eo}$	Eötvös number $\rho_{w}gd^{2}/\sigma_{aw}$	$[5,\,40]$	$\sigma_{aw}$
2	$\mathrm{Re}$	Reynolds number $\rho_{w}Ud/\eta_{w}$	$[20,\,70]$	$\eta_{w}$
3	$s$	normalized oil spreading coeff. $\Sigma_{o}/\sigma_{aw}$	$[0.02,\,0.50]$	$\sigma_{ao}$
4	$r_{ow}$	tension ratio $\sigma_{ow}/\sigma_{aw}$	$[0.30,\,0.70]$	$\sigma_{ow}$
5	$\rho_{o}/\rho_{w}$	oil-to-water density ratio	$[0.70,\,0.95]$	$\rho_{o}$
6	$\rho_{a}/\rho_{w}$	air-to-water density ratio	$[0.01,\,0.10]$	$\rho_{a},\ \eta_{a}$

Appendix

Table 10: The nine Sobol design variables and their sampling ranges. Parameters 1–3 control capillarity and wetting, 4–7 the material contrasts, 8–9 the initial bubble placement.

Figure 9: Ensemble diagnostics across all members, from Eqs. 49 and 50 . The response surfaces are smooth and non-bifurcating: every member follows the same qualitative sequence — rise, deform, breakthrough, entrain, detach.

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Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

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Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative

L^2

errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

Abstract

PDE Operator Learning

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