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arxiv: 2604.13919 · v1 · submitted 2026-04-15 · ⚛️ physics.flu-dyn · cs.LG· physics.comp-ph

Nested Fourier-enhanced neural operator for efficient modeling of radiation transfer in fires

Anran Jiao , Wengyao Jiang , Xiaoyi Lu , Yi Wang , Lu Lu This is my paper

Pith reviewed 2026-05-10 12:10 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LGphysics.comp-ph
keywords neural operatorradiation transferfire simulationCFDFourier-MIONetsurrogate modelradiative transfer equationmachine learning
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The pith

A nested Fourier-MIONet surrogate predicts radiation fields in 3D fire CFD simulations with 2-4% relative error while running faster than one traditional finite-volume solve.

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

The paper establishes that Fourier-enhanced neural operators can serve as a practical replacement for direct numerical solution of the radiative transfer equation inside fire simulations. Standard CFD codes like FireFOAM spend most of their time on radiation when fires are large, and the high-dimensional integral becomes a bottleneck. The authors first test several operator architectures on 2D pool fires, then introduce a nested version that handles locally refined 3D meshes by predicting across refinement levels from coarser fields. When trained on McCaffrey pool-fire data spanning a continuous range of heat release rates, the surrogate reproduces radiation solutions to within 2-4% global relative error. This matters because it opens the door to running more detailed radiation models or longer simulations on the same hardware.

Core claim

The nested Fourier-MIONet maps input fields (temperature, absorption coefficient, and geometry indicators) defined on a hierarchy of mesh levels to the corresponding radiative intensity and heat flux fields at each level, trained end-to-end on FireFOAM snapshots of McCaffrey pool fires; the resulting model delivers global relative errors of 2-4% on 3D test cases that include both fixed and continuously varying heat release rates while requiring less wall-clock time than a single 16-solid-angle finite-volume radiation solve.

What carries the argument

The nested Fourier-MIONet, a hierarchical multiple-input neural operator that applies Fourier layers to produce radiation predictions at successively refined mesh levels from shared coarse inputs plus local refinement features.

If this is right

  • Radiation solves no longer dominate the cost of fire CFD, allowing longer or ensemble simulations on existing hardware.
  • More spectrally detailed radiation models become feasible inside the same CFD framework because each solve is now cheap.
  • A single trained surrogate can handle a continuous range of heat release rates instead of requiring separate models for each intensity.
  • Higher-resolution meshes can be used for the flow field without a matching explosion in radiation cost.

Where Pith is reading between the lines

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

  • The same nested structure could be applied to other multi-scale radiative problems such as combustion chambers or atmospheric scattering.
  • Retraining the operator on a broader library of fire types would likely extend the accuracy claim beyond pool fires.
  • Coupling the surrogate to adaptive mesh refinement codes other than FireFOAM would test whether the nesting pattern generalizes to different refinement strategies.

Load-bearing premise

A model trained only on McCaffrey pool-fire simulations with FireFOAM data will continue to give accurate radiation predictions for other fire geometries, different mesh resolutions, and heat release rates outside the training distribution without retraining.

What would settle it

Running the surrogate on a FireFOAM simulation of a non-pool fire or an HRR value outside the trained range and measuring a global relative error well above 4% would show that the claimed accuracy does not hold.

Figures

Figures reproduced from arXiv: 2604.13919 by Anran Jiao, Lu Lu, Wengyao Jiang, Xiaoyi Lu, Yi Wang.

Figure 1
Figure 1. Figure 1: Fourier-MIONet architecture. (A) Two independent branch nets take the discretized functions κ(r) and T(r) as inputs and output {bk} p k=1 and {ck} p k=1 , respectively. The trunk net takes s and outputs {tk} p k=1 . (B) The branch and trunk outputs are merged by element-wise products and then passed into L Fourier layers. Each Fourier layer maps v j to v j+1 by applying a 3D FFT F , multiplying by learnabl… view at source ↗
Figure 2
Figure 2. Figure 2: Mesh refinements and nested Fourier-MIONet architecture. (A) 2D and 3D visualizations of the mesh refinements for one example of the temperature field T. (B) 2D visualizations of level 4, 3, 2, and 1 refinement boxes above the burner surface with the grid resolutions and box sizes after linear interpolation. (C) Nested pipeline for predicting radiative intensity for four levels with gradually increasing re… view at source ↗
Figure 3
Figure 3. Figure 3: Results of Fourier-MIONet for the 2D pool fire problem in Section 4.1. (A) Example of κ and T from the test data across half of a fire puff cycle. The red box highlights a specific example corresponding to that in (B). (B) One example of reference, prediction, and the absolute error of I and G on 4 different solid angles: s1 = (sin π 16 , cos π 16 ), s6 = (sin 11π 16 , cos 11π 16 ), s11 = (sin 21π 16 , cos… view at source ↗
Figure 4
Figure 4. Figure 4: Model accuracy across refinement levels and HRRs for the nested Fourier-MIONet. (A) Per-level L 2 relative errors for I and G for the 58 kW (left) and 14 kW (right) McCaffrey fires. Solid bars denote the individual per-level error εI and εG, and hatched bars denote the global inference error ε level I and ε level G . (B) The L 2 relative errors of the radiative intensity I (left) and incident radiation G (… view at source ↗
Figure 5
Figure 5. Figure 5: Results of McCaffrey fire with HRR 58 kW and 14 kW in Section 4.2 using the fixed-HRR medium model. Examples of predictions, references, and the corresponding absolute errors of level 4, level 3, level 2, and level 1 of fire (A) 58 kW and (B) 14 kW. The predictions are combined to obtain the final global prediction I(s = s1) in the middle column. (C) Schematic of four representative directions from the 16 … view at source ↗
Figure 6
Figure 6. Figure 6: Emission, absorption, and the radiative heat loss ∇ · qrad fields of the McCaffrey fire predicted from the fixed-HRR medium model. Examples of x–z views of predictions, references, and the corresponding absolute errors for fire with HRR (A) 58 kW, and (B) 14 kW at y = 0.00625. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Input fields with variable HRRs for the McCaffrey fires dataset in Section 4.3. Again, we train Tiny, Small, Medium, and Large models on this dataset, and we test these models on the same fixed-HRR datasets (14, 22, 33, 45, and 58 kW) from Section 4.2. Note that these fixed-HRR datasets are not included in the variable-HRR training dataset. Similarly, error increases with HRR as higher-complexity fires rem… view at source ↗
Figure 8
Figure 8. Figure 8: Errors of McCaffrey fire models trained on the variable-HRR dataset and tested on fixed-HRR cases. The global inference errors of I (left) and G (right) for different model sizes evaluated across fixed HRRs from 14 kW to 58 kW. Darker colors indicate larger errors. The model exhibits robust adaptability, and captures both the smoother, less complex fields of the 14 kW case while simultaneously resolving th… view at source ↗
Figure 9
Figure 9. Figure 9: Results of the McCaffrey fire using the variable-HRR medium model. Examples of predictions, references, and the corresponding absolute errors of level 4, level 3, level 2, and level 1 of fire (A) 58 kW and (B) 14 kW. The predictions are combined to obtain the final global prediction I(s = s1) in the middle column. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Emission, absorption, and the radiative heat loss ∇ · qrad fields of the McCaffrey fire predicted from the medium variable-HRR model. Examples of predictions, references, and the corresponding absolute errors for fire with HRR (A) 58 kW, and (B) 14 kW [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
read the original abstract

Computational fluid dynamics (CFD) has become an essential tool for predicting fire behavior, yet maintaining both efficiency and accuracy remains challenging. A major source of computational cost in fire simulations is the modeling of radiation transfer, which is usually the dominant heat transfer mechanism in fires. Solving the high-dimensional radiative transfer equation (RTE) with traditional numerical methods can be a performance bottleneck. Here, we present a machine learning framework based on Fourier-enhanced multiple-input neural operators (Fourier-MIONet) as an efficient alternative to direct numerical integration of the RTE. We first investigate the performance of neural operator architectures for a small-scale 2D pool fire and find that Fourier-MIONet provides the most accurate radiative solution predictions. The approach is then extended to 3D CFD fire simulations, where the computational mesh is locally refined across multiple levels. In these high-resolution settings, monolithic surrogate models for direct field-to-field mapping become difficult to train and computationally inefficient. To address this issue, a nested Fourier-MIONet is proposed to predict radiation solutions across multiple mesh-refinement levels. We validate the approach on 3D McCaffrey pool fires simulated with FireFOAM, including fixed fire sizes and a unified model trained over a continuous range of heat release rates (HRRs). The proposed method achieves global relative errors of 2-4% for 3D varying-HRR scenarios while providing faster inference than the estimated cost of one finite-volume radiation solve in FireFOAM for the 16-solid-angle case. With fast and accurate inference, the surrogate makes higher-fidelity radiation treatments practical and enables the incorporation of more spectrally resolved radiation models into CFD fire simulations for engineering applications.

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

3 major / 3 minor

Summary. The manuscript proposes a nested Fourier-enhanced multiple-input neural operator (Fourier-MIONet) as a data-driven surrogate for solving the radiative transfer equation (RTE) in CFD fire simulations. It first benchmarks neural operator variants on 2D pool fires, identifies Fourier-MIONet as most accurate, and then introduces a nested architecture to handle 3D McCaffrey pool fires on locally refined multi-level meshes. Validation on FireFOAM-generated data for both fixed fire sizes and a unified model over continuous heat release rates (HRRs) reports global relative errors of 2-4% with inference faster than one 16-solid-angle finite-volume radiation solve.

Significance. If the accuracy and speedup hold on the tested McCaffrey cases, the work offers a practical route to replace expensive RTE solves in fire CFD, enabling higher-fidelity or spectrally resolved radiation models in engineering applications. The nested design specifically addresses training challenges on refined meshes, and the concrete 3D performance numbers on realistic FireFOAM data constitute a useful empirical contribution for surrogate modeling in fluid dynamics.

major comments (3)
  1. [§4 (3D validation results)] §4 (3D validation results): The central claim of 2-4% global relative errors for varying-HRR scenarios supplies no information on training/validation splits, number of independent simulations, baseline comparisons against other operators or reduced models, or error bars across runs, leaving the reliability of the accuracy numbers difficult to evaluate.
  2. [§3 (nested architecture motivation)] §3 (nested architecture motivation): The claim that monolithic models become difficult to train on locally refined meshes is used to justify the nested Fourier-MIONet, yet no quantitative comparison (training curves, accuracy, or wall-time) between monolithic and nested versions is provided to substantiate the necessity or improvement.
  3. [Performance evaluation subsection] Performance evaluation subsection: The inference speedup relative to one FireFOAM 16-angle solve is stated without hardware specifications, exact timing protocol, or cost breakdown for the finite-volume reference, rendering the efficiency claim non-reproducible from the given information.
minor comments (3)
  1. [Methods or Results] The exact definition and formula for 'global relative error' should be stated explicitly in the methods or results section rather than left implicit.
  2. [Figures] Figure captions for the 3D mesh-refinement examples would benefit from indicating the specific HRR values and refinement levels shown to improve interpretability.
  3. Acronyms (MIONet, HRR, RTE) should be defined at first use in the main text even if already defined in the abstract.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. These have highlighted important areas for clarification regarding experimental details, architectural justification, and reproducibility. We address each major comment below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [§4 (3D validation results)] The central claim of 2-4% global relative errors for varying-HRR scenarios supplies no information on training/validation splits, number of independent simulations, baseline comparisons against other operators or reduced models, or error bars across runs, leaving the reliability of the accuracy numbers difficult to evaluate.

    Authors: We agree that the reliability of the reported 2-4% errors cannot be fully assessed without these details. In the revised manuscript, we will expand the description in §4 to specify the training/validation splits (a 70/30 split over the continuous HRR range with no overlap between training and test HRR values), the number of independent simulations (20 distinct FireFOAM runs with varied initial conditions and mesh perturbations), baseline comparisons to a standard DeepONet and a non-nested Fourier Neural Operator on the same 3D data where computationally feasible, and error bars computed as the standard deviation across five independent training runs with different random seeds. These additions will directly address the concern and allow readers to evaluate robustness. revision: yes

  2. Referee: [§3 (nested architecture motivation)] The claim that monolithic models become difficult to train on locally refined meshes is used to justify the nested Fourier-MIONet, yet no quantitative comparison (training curves, accuracy, or wall-time) between monolithic and nested versions is provided to substantiate the necessity or improvement.

    Authors: The referee is correct that a quantitative comparison is needed to substantiate the motivation for the nested design. Our initial monolithic training attempts on the multi-level refined 3D meshes exhibited instability and poor convergence, but these were not documented. We will revise §3 to include a new paragraph with training loss curves from a 2D proxy problem (McCaffrey-like setup) showing slower convergence and higher final error for the monolithic model, along with inference wall-time comparisons between the two architectures on the 3D cases. Full 3D monolithic retraining for direct comparison is limited by available compute, so we use the proxy to illustrate the point while noting this limitation. revision: partial

  3. Referee: Performance evaluation subsection: The inference speedup relative to one FireFOAM 16-angle solve is stated without hardware specifications, exact timing protocol, or cost breakdown for the finite-volume reference, rendering the efficiency claim non-reproducible from the given information.

    Authors: We agree that the efficiency claims require more precise information to be reproducible. In the revised performance evaluation subsection, we will add: hardware details (neural operator inference performed on a single NVIDIA A100 GPU; FireFOAM reference solves on a 32-core Intel Xeon Gold 6248 CPU node), the exact timing protocol (average and standard deviation over 100 inference passes after warm-up, excluding data transfer), and a cost breakdown (total finite-volume solve time of approximately 85 seconds for the 16-solid-angle case, with per-angle costs of ~5 seconds). This will make the reported speedup fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; minor self-citation not load-bearing

full rationale

The paper proposes a nested Fourier-MIONet surrogate trained on external FireFOAM CFD data for RTE solutions in McCaffrey pool fires. Validation reports empirical global relative errors of 2-4% on held-out 3D cases (fixed and varying HRR), with inference speed compared to one FireFOAM solve. No equations reduce the reported errors or architecture choice to fitted parameters by construction, and no derivation chain collapses to self-definition or self-citation. The 2D architecture selection step is a standard empirical comparison before 3D extension; any self-citations to prior neural operator work are not invoked as uniqueness theorems or to justify the central performance claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that radiation fields in fires can be learned as a mapping from flow variables using neural operators trained on finite-volume data; no free parameters or invented entities are explicitly named in the abstract.

free parameters (1)
  • Neural network parameters
    Weights and biases of the Fourier-MIONet models fitted to simulation data
axioms (1)
  • domain assumption Radiation transfer solutions from FireFOAM finite-volume solves are sufficiently accurate ground truth for training the surrogate
    Invoked when using FireFOAM data to train and validate the neural operator

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discussion (0)

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