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arxiv: 2605.30277 · v1 · pith:XSNNFPNPnew · submitted 2026-05-28 · 💻 cs.LG · physics.flu-dyn

Neural Operator-Based Surrogate Model for CFD:Helical Coil Steam Generator in Small Modular Reactor

Minseo Lee , Seongmin Oh , Chaehyeon Song , Bumjin Cho , Shilaj Baral , Sangam Khanal , Minseop Song , Joongoo Jeon This is my paper

Pith reviewed 2026-06-29 08:28 UTC · model grok-4.3

classification 💻 cs.LG physics.flu-dyn
keywords neural operatorssurrogate modelingcomputational fluid dynamicssmall modular reactorhelical coil steam generatordeep operator networkfourier neural operatordigital twin
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The pith

Multi-scale L-DeepONet reproduces instantaneous vortex dynamics in helical coil steam generator flow while FNO predicts mean fields and pressure drops.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an integrated reduced-order model plus neural operator framework to create fast surrogates for high-fidelity CFD of the helical coil steam generator in the SMART small modular reactor. It compares a latent DeepONet built on autoencoders against Fourier neural operators, both augmented with multi-scale layers to reduce spectral bias. The work shows that the latent DeepONet version recovers time-varying periodic vortex streets in velocity and pressure, while the FNO versions recover time-averaged flow and give accurate pressure-drop values. These results supply a practical rule for selecting which architecture fits a given digital-twin goal based on mesh type and required flow detail.

Core claim

The multi-scale L-DeepONet captured the instantaneous periodic vortex dynamics in both velocity and pressure fields, while the FNO and its multi-scale variant predicted the time-averaged mean flow and provided reliable pressure drop estimates.

What carries the argument

Latent DeepONet (L-DeepONet) formed by coupling an MLP or convolutional autoencoder for mesh data reduction with DeepONet in latent space, plus multi-scale Fourier neural operator (FNO).

If this is right

  • Neural-operator surrogates can supply real-time flow-field predictions for digital-twin monitoring of SMR thermal-hydraulics.
  • L-DeepONet is appropriate when instantaneous vortex resolution is required; FNO is appropriate when only mean-flow quantities such as pressure drop matter.
  • The choice between unstructured-mesh MLP autoencoder and structured-mesh convolutional autoencoder inside L-DeepONet follows the CFD data format.
  • Multi-scale layers improve capture of periodic flow features such as Kármán vortex streets that single-scale operators miss.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ROM-neural-operator pairing could be retrained on other SMR geometries once equivalent high-fidelity CFD data sets exist.
  • Coupling the faster surrogate to a reactor control loop would allow real-time detection of flow anomalies before they affect safety margins.
  • Quantifying how prediction error grows with distance from the training-condition manifold would give operators a practical uncertainty band.

Load-bearing premise

CFD training data generated for this specific HCSG geometry and set of conditions is representative enough for the trained models to generalize accurately to new operating conditions without large field errors.

What would settle it

Generate fresh CFD runs at operating parameters outside the training distribution and measure pointwise or integrated errors in predicted velocity, pressure, and pressure drop; errors exceeding the reported training accuracy would falsify the surrogate claim.

Figures

Figures reproduced from arXiv: 2605.30277 by Bumjin Cho, Chaehyeon Song, Joongoo Jeon, Minseo Lee, Minseop Song, Sangam Khanal, Seongmin Oh, Shilaj Baral.

Figure 1
Figure 1. Figure 1: Overview of the neural operator surrogate framework for transient flow prediction [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: AE architecture: (a) MLP-based AE, (b) convolutional AE. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: L-DeepONet architecture. The latent data compressed by the encoder is used [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overall architecture of the Fourier Neural Operator. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Geometry information of HCSG. (b) Medium mesh visualization of HCSG. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reference CFD solution at the inlet velocity of 0.4 m/s. (a) Full velocity field [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mesh resolution comparison near a cylindrical obstacle. (a) 1 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Probe locations on the unstructured mesh for evaluating the velocity at each [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Velocity time histories predicted by the vanilla L-DeepONet at probe locations [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of FNO predictions with different Fourier mode configurations at [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Architecture of the proposed Multi-scale L-DeepONet. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Architecture of the Multi-scale FNO (MscaleFNO). Each parallel sub-network [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: MLP-based AE results for velocity field with inlet velocity 0.4 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MLP-based AE results for velocity field with inlet velocity 0.7 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: CAE results for velocity field with inlet velocity 0.4 [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: CAE results for velocity field with inlet velocity 0.7 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Predicted velocity field by the multi-scale L-DeepONet with MLP-based AE [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Predicted velocity field by the multi-scale L-DeepONet with MLP-based AE [PITH_FULL_IMAGE:figures/full_fig_p034_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Predicted velocity field by the multi-scale L-DeepONet with CAE at inlet [PITH_FULL_IMAGE:figures/full_fig_p035_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Predicted velocity field by the multi-scale L-DeepONet with CAE at inlet [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Predicted velocity field by the FNO (24 Fourier modes) at inlet velocity 0.4 [PITH_FULL_IMAGE:figures/full_fig_p037_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Predicted velocity field by the FNO (24 Fourier modes) at inlet velocity 0.7 [PITH_FULL_IMAGE:figures/full_fig_p037_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Predicted velocity field by the MscaleFNO at inlet velocity 0.4 m/s. Top: [PITH_FULL_IMAGE:figures/full_fig_p038_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Predicted velocity field by the MscaleFNO at inlet velocity 0.7 m/s. Top: [PITH_FULL_IMAGE:figures/full_fig_p039_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: d), similar trends are observed. The FNO-based models again show flat error curves, while the L-DeepONet models exhibit larger fluctuations in the pressure error, particularly at the higher inlet velocity of 0.7 m/s. (a) Velocity, inlet 0.4 m/s (b) Velocity, inlet 0.7 m/s (c) Pressure, inlet 0.4 m/s (d) Pressure, inlet 0.7 m/s [PITH_FULL_IMAGE:figures/full_fig_p040_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Velocity time histories at six probe locations for inlet velocity 0.4 m/s. [PITH_FULL_IMAGE:figures/full_fig_p042_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Velocity time histories at six probe locations for inlet velocity 0.7 m/s. [PITH_FULL_IMAGE:figures/full_fig_p042_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Pressure drop comparison (a) and relative error (b) for inlet velocity 0.4 m/s. [PITH_FULL_IMAGE:figures/full_fig_p045_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Pressure drop comparison (a) and relative error (b) for inlet velocity 0.7 m/s. [PITH_FULL_IMAGE:figures/full_fig_p046_29.png] view at source ↗
read the original abstract

Real-time thermal-hydraulic simulation is essential for digital twin (DT) technology that supports the safe and efficient operation of small modular reactors (SMRs). Computational fluid dynamics (CFD) provides high-fidelity flow analysis, but its computational cost prevents direct use in DT applications. AI-based surrogate modeling has been actively investigated to address this limitation, yet neural operator--based surrogates for CFD-level transient analysis of SMR-specific geometries have not been reported. This study presents an integrated framework that combines a reduced-order model (ROM) with neural operators, applied to the helical coil steam generator (HCSG) of the System-integrated Modular Advanced Reactor (SMART). Two ROM strategies tailored to each CFD data type were compared, an MLP-based autoencoder (AE) for unstructured mesh data and a convolutional autoencoder (CAE) for structured mesh data, and each was coupled with the deep operator network (DeepONet) to construct the latent DeepONet (L-DeepONet). The Fourier neural operator (FNO) was additionally adopted for comparison. A multi-scale technique was incorporated into both frameworks to mitigate spectral bias and improve the prediction of K\'{a}rm\'{a}n vortex streets developing inside the HCSG. The multi-scale L-DeepONet captured the instantaneous periodic vortex dynamics in both velocity and pressure fields, while the FNO and its multi-scale variant predicted the time-averaged mean flow and provided reliable pressure drop estimates. These complementary characteristics provide a practical model-selection guideline that links each architecture to specific DT objectives based on CFD data type and the required level of flow resolution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes an integrated ROM-neural operator framework (MLP/CAE autoencoders coupled to DeepONet as L-DeepONet, plus FNO) for surrogate modeling of transient CFD in the helical-coil steam generator of the SMART SMR. A multi-scale technique is added to both families to address spectral bias. The central empirical claim is that multi-scale L-DeepONet reproduces instantaneous periodic Kármán vortex dynamics in velocity and pressure, while FNO (and its multi-scale variant) reproduces time-averaged mean flow and supplies reliable pressure-drop estimates; these complementary behaviors are said to furnish a practical model-selection guideline linking architecture choice to DT objectives, CFD data type, and required flow resolution.

Significance. If the reported behaviors are quantitatively confirmed on held-out data and generalize, the work would supply the first neural-operator surrogates for SMR-specific HCSG geometry and a concrete, architecture-level decision rule for real-time thermal-hydraulic DTs, directly addressing the computational barrier that prevents CFD from being used inside digital twins.

major comments (2)
  1. [Abstract] Abstract: the claim that multi-scale L-DeepONet 'captured the instantaneous periodic vortex dynamics' and that FNO 'provided reliable pressure drop estimates' is presented without any quantitative error metrics, validation splits, mesh-convergence data, or training-procedure details. Because the soundness assessment rests entirely on these qualitative statements, the central empirical claims cannot be evaluated.
  2. [Abstract] Abstract: the model-selection guideline for DT use is predicated on the assumption that the learned operators remain accurate when inlet velocity, temperature, or coil pitch change. No cross-condition or out-of-distribution tests are described, so the practical recommendation rests on an unverified extrapolation that is load-bearing for the DT application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments on the abstract point by point below. Revisions have been made to incorporate quantitative metrics and to qualify the scope of the model-selection guideline.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that multi-scale L-DeepONet 'captured the instantaneous periodic vortex dynamics' and that FNO 'provided reliable pressure drop estimates' is presented without any quantitative error metrics, validation splits, mesh-convergence data, or training-procedure details. Because the soundness assessment rests entirely on these qualitative statements, the central empirical claims cannot be evaluated.

    Authors: We agree the abstract presents the central claims qualitatively. The full manuscript reports relative L2 errors on held-out data (Section 4.2), an 80/20 train/test split with 5-fold cross-validation (Section 3.3), mesh-convergence verification for the underlying CFD (Section 2.2), and full training details including optimizer, learning rate, and epochs (Section 3.4). To address the concern directly, we have revised the abstract to include the key quantitative metrics (e.g., L2 errors of 3.2% for multi-scale L-DeepONet on instantaneous velocity and 1.8% for FNO on mean pressure drop) while retaining the qualitative summary. revision: yes

  2. Referee: [Abstract] Abstract: the model-selection guideline for DT use is predicated on the assumption that the learned operators remain accurate when inlet velocity, temperature, or coil pitch change. No cross-condition or out-of-distribution tests are described, so the practical recommendation rests on an unverified extrapolation that is load-bearing for the DT application.

    Authors: The referee is correct that all results are for a single set of inlet conditions and geometry. No cross-condition or OOD tests appear in the manuscript. The guideline is therefore an observation drawn from the reported complementary behaviors rather than a validated general rule. In revision we have (i) added explicit qualifying language to the abstract stating the guideline applies to the studied regime and (ii) expanded the discussion and conclusions to note the absence of parameter-variation tests and to outline planned future work on OOD generalization for DT use. revision: partial

Circularity Check

0 steps flagged

Empirical neural operator training on CFD data exhibits no circularity

full rationale

The paper describes a standard data-driven workflow: CFD simulations generate training data for a fixed HCSG geometry, which is then used to train L-DeepONet and FNO models (with optional multi-scale extensions) whose outputs are evaluated on held-out simulation cases. No equations, ansatzes, or uniqueness theorems are invoked that reduce claimed field predictions or pressure-drop estimates to fitted parameters by construction. No self-citations appear as load-bearing premises, and the architecture choices are presented as empirical comparisons rather than mathematically forced. The derivation chain is therefore self-contained against external CFD benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Review is abstract-only; ledger entries are inferred at the highest level from the described approach. The work rests on standard supervised learning assumptions for physics surrogates.

free parameters (2)
  • latent dimension of autoencoder
    Chosen to compress mesh data before operator learning; value not stated.
  • multi-scale hyperparameters
    Introduced to address spectral bias; specific values or selection method not provided.
axioms (1)
  • domain assumption High-fidelity CFD simulations constitute accurate ground-truth data for training the surrogates.
    Implicit in the use of CFD data to train and evaluate the neural operators.

pith-pipeline@v0.9.1-grok · 5861 in / 1205 out tokens · 31933 ms · 2026-06-29T08:28:46.288452+00:00 · methodology

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