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

Anran Jiao; Wengyao Jiang; Xiaoyi Lu; Yi Wang; Lu Lu

physics.flu-dynApr 15, 2026

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

Authors: Anran Jiao, Wengyao Jiang, Xiaoyi Lu, Yi Wang, Lu Lu

Organizations: Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA · Department of Biostatistics, Yale University, New Haven, CT 06511, USA · FM, Research Division, Norwood, MA 02062, USA · Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06511, USA

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.

Figures & tables

Figure 1: Fourier-MIONet architecture. ( A ) Two independent branch nets take the discretized functions $\bm{\kappa}(\mathbf{r})$ and $\mathbf{T}(\mathbf{r})$ as inputs and output $\{b_{k}\}_{k=1}^{p}$ and $\{c_{k}\}_{k=1}^{p}$ , respectively. The trunk net takes $\mathbf{s}$ and outputs $\{t_{k}\}_{k=1}^{p}$ . ( 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 $\mathcal{F}$ , multiplying by learnable weights $\mathcal{R}_{j}$ in the Fourier space, and transforming back with $\mathcal{F}^{-1}$ . A residual connection with a weight matrix $W_{j}$ is added.

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 resolution. Each level, starting from level 4, uses outputs from the previous coarser level on top of the absorption coefficient, temperature, and spatial directions (black arrows). The dashed arrows represent the deconvolution layers. The final global solution $I(\mathbf{r},\mathbf{s})$ for the entire domain is assembled from the individual predictions by selecting the finest available resolution for each spatial node.

	$\epsilon_{I}$	$\text{SSIM}_{I}$	$\epsilon_{G}$	$\text{SSIM}_{G}$	#Params
PCA-MIONet	7.86%	0.930	4.73%	0.978	1586178
PCA-MIONet-KAN	6.71%	0.950	4.16%	0.983	1576564
MIONet-KAN	7.56%	0.939	4.61%	0.979	1587232
Fourier-MIONet	3.70%	0.982	1.93%	0.996	1583649

Table 1: Comparison of PCA-MIONet, PCA-MIONet-KAN, MIONet-KAN, and Fourier-MIONet on the 2D pool fire problem.

Figure 3: Results of Fourier-MIONet for the 2D pool fire problem in Section 4.1 . ( A ) Example of $\kappa$ 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: $s_{1}=(\sin{\frac{\pi}{16}},\cos{\frac{\pi}{16}})$ , $s_{6}=(\sin{\frac{11\pi}{16}},\cos{\frac{11\pi}{16}})$ , $s_{11}=(\sin{\frac{21\pi}{16}},\cos{\frac{21\pi}{16}})$ , and $s_{15}=(\sin{\frac{29\pi}{16}},\cos{\frac{29\pi}{16}})$ .

Fire	Model	Individual error $\varepsilon^{\text{level}}_{I}$				$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
Fire	Model	Level 4	Level 3	Level 2	Level 1	$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
58 kW	Tiny	10.50%	15.80%	9.95%	9.77%	5.82%	2.81%	0.15%	0.15%	0.14 $~\mathrm{kW}$	0.01 $~\mathrm{kW}$
	Small	7.68%	10.54%	5.78%	6.03%	3.89%	2.00%	0.04%	0.03%	0.11 $~\mathrm{kW}$
	Medium	5.39%	6.45%	3.73%	2.79%	2.82%	1.55%	0.02%	0.02%	0.00 $~\mathrm{kW}$
	Large	2.63%	2.57%	2.22%	2.11%	1.38%	0.55%	0.05%	0.06%	0.01 $~\mathrm{kW}$
14 kW	Tiny	3.69%	7.47%	6.20%	6.16%	2.19%	0.87%	0.19%	0.19%	0.35 $~\mathrm{kW}$	0.45 $~\mathrm{kW}$

Table 2: The global testing performance of the nested Fourier-MIONet for the 3D McCaffrey 58 kW and 14 kW fire. We report the individual per-level errors and the global errors obtained during nested inference. The reference radiative fractions

\chi_{R}

are 21.26% and 17.83% for the 58 kW and 14 kW fires, respectively.

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 $58~\mathrm{kW}$ (left) and $14~\mathrm{kW}$ (right) McCaffrey fires. Solid bars denote the individual per-level error $\varepsilon_{I}$ and $\varepsilon_{G}$ , and hatched bars denote the global inference error $\varepsilon_{I}^{\text{level}}$ and $\varepsilon_{G}^{\text{level}}$ . ( B ) The $L^{2}$ relative errors of the radiative intensity $I$ (left) and incident radiation $G$ (right) versus the model size for 14, 22, 33, 45, and $58~\mathrm{kW}$ cases during global inference.

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 ) $58~\mathrm{kW}$ and ( B ) $14~\mathrm{kW}$ . The predictions are combined to obtain the final global prediction $I(\mathbf{s}=s_{1})$ in the middle column. ( C ) Schematic of four representative directions from the 16 directions we sampled on the unit sphere. They are defined by the corresponding polar and azimuthal angles $s_{k}=(\theta_{i},\phi_{j})$ , with $k=8i+j$ , where $\theta_{i}=\frac{(2i+1)\Delta\theta}{2},\ i=0,1$ , and $\phi_{j}=\frac{(2j+1)\Delta\phi}{2},\ j=0,\ldots,7$ .

Figure 6: Emission, absorption, and the radiative heat loss $\nabla\cdot q_{\mathrm{rad}}$ 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 ) $58~\mathrm{kW}$ , and ( B ) $14~\mathrm{kW}$ at $y$ = 0.00625.

Figure 7: Input fields with variable HRRs for the McCaffrey fires dataset in Section 4.3 .

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.

Model	HRR	Individual error $\varepsilon^{\text{level}}_{I}$				$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
Model	HRR	Level 4	Level 3	Level 2	Level 1	$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
Tiny	$14~\mathrm{kW}$	4.55%	6.88%	8.85%	8.70%	3.06%	1.63%	1.80%	1.80%	0.42 $~\mathrm{kW}$	0.45 $~\mathrm{kW}$
	$22~\mathrm{kW}$	5.69%	8.04%	7.58%	7.32%	3.10%	1.55%	0.38%	0.38%	0.63 $~\mathrm{kW}$	0.45 $~\mathrm{kW}$
	$33~\mathrm{kW}$	7.80%	9.48%	8.38%	8.88%	4.06%	2.24%	1.19%	1.19%	0.77 $~\mathrm{kW}$	0.37 $~\mathrm{kW}$
	$45~\mathrm{kW}$	9.29%	10.67%	9.61%	10.27%	5.00%	2.77%	1.15%	1.15%	0.59 $~\mathrm{kW}$	0.15 $~\mathrm{kW}$
	$58~\mathrm{kW}$	10.40%	11.63%	10.56%	11.53%	5.92%	3.28%	0.93%	0.92%	0.31 $~\mathrm{kW}$	0.01 $~\mathrm{kW}$

Table 3: 3D McCaffrey fire trained on variable HRRs and tested on fixed HRR fires. We summarize the individual per-level errors and the global errors obtained during nested inference for each HRR.

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 ) $58~\mathrm{kW}$ and ( B ) $14~\mathrm{kW}$ . The predictions are combined to obtain the final global prediction $I(\mathbf{s}=s_{1})$ in the middle column.

Figure 10: Emission, absorption, and the radiative heat loss $\nabla\cdot q_{\mathrm{rad}}$ 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 ) $58~\mathrm{kW}$ , and ( B ) $14~\mathrm{kW}$ .

Model	Individual inference time				Total inference time	fvDOM runtime (CPU, 48 cores)
Model	Level 4	Level 3	Level 2	Level 1	Total inference time	fvDOM runtime (CPU, 48 cores)
Tiny	0.0027	0.0034	0.0023	0.0073	0.0157	0.10
Small	0.0038	0.0021	0.0040	0.0120	0.0219
Medium	0.0039	0.0025	0.0050	0.0147	0.0261
Large	0.0043	0.0047	0.0094	0.0259	0.0443

Table 4: Inference time (s) of the nested Fourier-MIONet on the 3D McCaffrey fire with variable HRRs, compared with fvDOM CPU runtime.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Interpolation	Level	$I$		$G$
Interpolation	Level	$\varepsilon_{I}$	SSIM I	$\varepsilon_{G}$	SSIM G
Linear	4	5.39%	0.995	2.24%	0.991
	3	4.02%	0.997	1.54%	0.998
	2	3.15%	0.997	1.36%	0.999
	1	2.39%	0.994	0.99%	0.999
Nearest neighbor	4	5.51%	0.995	2.30%	0.990

Appendix

Table 5: Comparison of linear versus nearest-neighbor interpolation on 3D McCaffrey

58~\mathrm{kW}

fire with per-level testing performance. The top panel reports per-level

L^{2}

relative errors and SSIM values for radiative intensity

I

and incident radiation

G

during training of the medium model. The lower panel summarizes the individual per-level errors and the global errors obtained during nested inference.

Figure 11: Illustration and comparison of data preprocessing approaches. ( A ) Comparison of the nearest neighbor interpolation approach (top path) and the linear interpolation approach (bottom path) on the raw mesh with a $T$ example at level 2. The nearest neighbor interpolation approach performs a direct point lookup based on minimal Euclidean distance and assigns the corresponding values to the target grid locations. The linear interpolation approach utilizes barycentric weighting via Delaunay triangulation and produces smoother and more accurate fields. ( B ) Reference, global predictions, and absolute errors of $I(\mathbf{s}=s_{1})$ obtained using the two preprocessing approaches for direct comparison.

Level		Operation	Output Shape
Level 4	Branch net 1	Linear(4, W), Permute	( $C$ , $W$ , 30, 30, 30)
	Branch net 2	Linear(4, W), Permute	( $C$ , $W$ , 30, 30, 30)
	Trunk net	FNN [2, $W$ , $W$ , $W$ ]	(16, $W$ )
	Branch merger	Point-wise addition	( $C$ , $W$ , 30, 30, 30)
	Branch-trunk merger	Point-wise multiplication	( $C$ , 16, $W$ , 30, 30, 30)
	Merge net	$N$ Fourier layers	( $C\times 16$ , $W$ , 30, 30, 30)

Appendix

Table 6: Architecture of the nested Fourier-MIONet across all four levels.

C

denotes the batch size,

W

the network width, and

N

the number of Fourier layers.

Model	Level	Width	Modes		# Fourier layers	# Params
Model	Level	Width	1&2	3	# Fourier layers	# Params
Tiny	4	16	4	4	1	67713
	3	16	4	4	1	176326
	2	16	4	4	1	127211
	1	16	4	4	1	131182
Small	4	24	8	4	2	1184857

Appendix

Table 7: Model and per-level configuration of the nested Fourier-MIONet used for the 3D McCaffrey fires. Four model sizes are considered: Tiny, Small, Medium, and Large, which differ in the network width (“Width”) of the branch, trunk, and Fourier decoder networks, the number of retained Fourier modes (“Modes”) in the three spatial dimensions, and the number of Fourier layers. The architecture and hyperparameters are shared across all sizes of fire (14–58 kW).

Fire	Model	Individual error $\varepsilon^{\text{level}}_{I}$				$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
Fire	Model	Level 4	Level 3	Level 2	Level 1	$\varepsilon^{\text{global}}_{I}$	$\varepsilon^{\text{global}}_{G}$	$\varepsilon_{\dot{Q}_{out}}$	$\varepsilon_{\chi_{R}}$	$\mathcal{R}_{E}^{pred}$	$\mathcal{R}_{E}^{ref}$
22 kW	Tiny	5.52%	10.74%	7.26%	6.85%	2.99%	1.35%	0.19%	0.19%	0.38 $~\mathrm{kW}$	0.45 $~\mathrm{kW}$
	Small	4.26%	8.28%	4.95%	4.86%	2.03%	1.14%	0.15%	0.15%	0.42 $~\mathrm{kW}$
	Medium	3.69%	6.15%	3.91%	2.81%	1.64%	0.98%	0.13%	0.13%	0.44 $~\mathrm{kW}$
	Large	2.43%	2.94%	2.25%	1.43%	1.04%	0.60%	0.03%	0.04%	0.44 $~\mathrm{kW}$
33 kW	Tiny	7.93%	13.93%	8.78%	8.40%	4.28%	2.02%	0.16%	0.16%	0.20 $~\mathrm{kW}$	0.37 $~\mathrm{kW}$

Appendix

Table 8: Per-level and global testing performance of the nested Fourier-MIONet for the 3D McCaffrey 22, 33, and 45 kW fires. We report the individual per-level errors and the global errors obtained during nested inference. The reference radiative fractions

\chi_{R}

are 19.40%, 20.91%, and 21.76% for the 22, 33, and 45 kW fires, respectively.

Radiative intensity $I$
Model	Level	14 kW		22 kW		33 kW		45 kW		58 kW
Model	Level	$\varepsilon_{I}$	SSIM I	$\varepsilon_{I}$	SSIM I	$\varepsilon_{I}$	SSIM I	$\varepsilon_{I}$	SSIM I	$\varepsilon_{I}$	SSIM I
Tiny	4	3.69%	0.996	5.52%	0.994	8.76%	0.921	9.75%	0.928	10.50%	0.986
	3	5.08%	0.995	6.28%	0.994	9.01%	0.898	9.54%	0.908	10.32%	0.983
	2	5.24%	0.993	5.24%	0.993	8.41%	0.816	9.88%	0.795	9.48%	0.972
	1	5.52%	0.986	6.16%	0.981	7.53%	0.966	8.60%	0.953	9.05%	0.949

Appendix

Table 9: Per-level testing

L^{2}

relative errors and SSIM values for radiative intensity

I

and incident radiation

G

during training on McCaffrey fires of 14, 22, 33, 45, and 58 kW. The upper block reports results for

I

, and the lower block reports results for

G

. Model configurations are provided in Table 7 .

Model	Level	$I$		$G$
Model	Level	$\varepsilon_{I}$	SSIM I	$\varepsilon_{G}$	SSIM G
Tiny	4	8.16%	0.989	3.54%	0.974
	3	9.59%	0.986	3.95%	0.988
	2	8.94%	0.970	4.01%	0.986
	1	8.12%	0.925	4.33%	0.962
Small	4	7.66%	0.993	3.17%	0.985

Appendix

Table 10: Per-level testing

L^{2}

relative errors and SSIM values for radiative intensity

I

and incident radiation

G

when training the entire nested Fourier-MIONet on variable HRRs.

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