A panoramic aerodynamic performance prediction method for turbomachinery cascades using transformer-enhanced neural operator

Qineng Wang; Zhendong Guo; Liming Song; Tianyuan Liu

cs.LGSep 13, 2026

A panoramic aerodynamic performance prediction method for turbomachinery cascades using transformer-enhanced neural operator

Authors: Qineng Wang, Zhendong Guo, Liming Song, Tianyuan Liu

Organizations: Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an, 710049, China · ENN Science and Technology Development China Co., Ltd., Langfang, 065001, China · Hebei Key Laboratory of Compact Fusion, Langfang, 065001, China

Abstract

To enable flexible and rapid aerodynamic performance evaluation in turbomachinery design, this paper proposes a panoramic performance prediction framework. Unlike most previous prediction models that directly predict the objective functions of interest, our approach first predicts the basic parameters of the Navier-Stokes equations, such as temperature, pressure, and density. Utilizing these basic physical quantities, it subsequently predicts key performance parameters of the turbine stage meridian plane. By adopting this methodology, our proposed panoramic performance prediction framework functions similarly to a CFD simulator, capable of predicting various objective of interest to the designers. To enhance prediction accuracy, a transformer-enhanced neural operator (TNO) is introduced within this framework. Using the Rotor 37 blades as a reference, the proposed TNO is trained to predict the performance of a transonic compressor blade in the meridian plane. The TNO can accurately predict total quantities such as isentropic efficiency, mass flow, and distributions of total pressure ratio. Remarkably, the prediction error of TNO is observed to be smaller than that of state-of-the-art deep learning operators such as the FNO and DeepONet. Furthermore, the TNO is applied to downstream tasks, including sensitivity analysis and optimization of various objective functions. The results confirm that the TNO can operate almost like a CFD simulator, while reducing the computational cost of downstream tasks by four orders of magnitude. The effectiveness and reliability of the proposed TNO for solving different kinds of downstream tasks have been well demonstrated.

Figures & tables

Condition name	Value
equivalent rotational speed[rpm]	17188.7
number of rotor blades	36
rotor blade aspect ratio	1.19
tip clearance gap[mm]	0.356
inlet total temperature[K]	288.15
inlet total pressure[Pa]	101325

Table 1: Design conditions of the Rotor 37 blade

Panel of Figure 1 — Figure 1: The geometry and mesh illustration of Rotor 37 blade

Geometric definition	Variable index	Range/ $C_{\mathrm{axial}}$
Leading edge	$x_{1},x_{6},x_{11},x_{16},x_{21}$	$[-0.6\%,0.6\%]$
Near-leading edge	$x_{2},x_{7},x_{12},x_{17},x_{22}$	$[-1.2\%,1.2\%]$
Intermediate	$x_{3},x_{8},x_{13},x_{18},x_{23}$	$[-1.5\%,1.5\%]$
Near-trailing edge	$x_{4},x_{9},x_{14},x_{19},x_{24}$	$[-1.2\%,1.2\%]$
Trailing edge	$x_{5},x_{10},x_{15},x_{20},x_{25}$	$[-0.6\%,0.6\%]$
Circumferential translation	$x_{26}$	$[-15\%,15\%]$

Table 2: Design variables of the Rotor 37 blade. Displacement ranges are multiples of the axial chord

C_{\mathrm{axial}}

Panel of Figure 2 — Figure 2: The comparison of feature curves of CFD simulation and experiment result

Figure 4: The network structure of the TNO network

Figure 5: The structure diagram of Galerkin attention block

Figure 6: The panoramic flow field and performance prediction framework

Network	MLP	UNet [ 35 ]	deepONet [ 23 ]	FNO [ 24 ]	TNO
parameter number $[\times 10^{3}]$	5731	3418	302	436	154
forward memory $[\text{MB}]$	768.0	624.5	822.5	558.9	147.13
training time $[\text{min}]$	19.98	114.0	52.9	106.5	40.5
predicting time $[\text{ms}]$	2.198	5.075	0.970	1.448	0.915

Table 3: The comparison of different network model establishment and training details

Figure 7: The convergence curves of loss functions of different networks

	$p$	$t$	$v^{2}$	$w^{2}$	$\rho v_{m}$	average
MLP	0.624%	0.245%	1.323%	0.954%	0.956%	0.820%
UNet	0.723%	0.229%	1.342%	0.852%	0.811%	0.792%
deepONet	1.244%	0.609%	2.894%	2.436%	2.257%	1.888%
FNO	0.472%	0.166%	1.054%	0.624%	0.546%	0.572%
TNO	0.060%	0.024%	0.153%	0.087%	0.091%	0.083%

Table 4: The related error of the predicted basic physical fields

Figure 8: Comparison of TNO network predicted and CFD results of basic flow fields for the case valid-A

Fields name	Symbol	Expression formula
absolute total temperature	$t^{*}_{v}$	$t^{*}_{v}=t+\frac{v^{2}}{2C_{p}}$
relative total temperature	$t^{*}_{w}$	$t^{*}_{w}=t+\frac{w^{2}}{2C_{p}}$
absolute total pressure	$p^{*}_{v}$	$p^{}_{v}=p(t^{}_{v}/t)^{\kappa/\kappa-1}$
relative total pressure	$p^{*}_{w}$	$p^{}_{w}=p(t^{}_{w}/t)^{\kappa/\kappa-1}$
static entropy	${H}$	${H}=\frac{Cp}{\kappa}\log\left(\left(\frac{p}{p_{\text{ref}}}\right)^{(1-\kappa)}\left(\frac{t}{t_{\text{ref}}}\right)^{\kappa}\right)$

Table 5: Explicit calculation method of various derived fields.

Figure 9: Compare of predicted and CFD results of derived flow fields for case valid-A

Performance name	Symbol	Expression formula
isentropic efficiency	$\eta_{is}$	$\eta_{is}=({\frac{p^{}_{v2}}{p^{}_{v1}}^{\frac{\kappa-1}{\kappa}}-1})/({\frac{t^{}_{v2}}{t^{}_{v1}}-1})$
polytropic efficiency	$\eta_{pol}$	$\eta_{pol}=\frac{R}{C_{p}}\cdot{\ln\left(\frac{p^{}_{v2}}{p^{}_{v1}}\right)}/{\ln\left(\frac{t^{}_{t2}}{t^{}_{v1}}\right)}$
pressure ratio	$\pi$	$\pi=p^{}_{v2}/p^{}_{v1}$
temperature ratio	$\theta$	$\theta=t^{}_{v2}/t^{}_{v1}$
mass flow	$\dot{m}$	$\dot{m}=\rho v_{m}\cdot\pi(r_{\text{s}}^{2}-r_{\text{h}}^{2})$
pressure loss coefficient	$\omega$	${\omega}=({p_{w2}^{}-p_{w1}^{}})/({p_{w1}^{*}-p_{1}})$

Table 6: Explicit calculation method of various performance metrics

Figure 10: Comparison between global performance values from CFD and the networks' predictions. The inset bands are $\pm 0.5\%$ . Reproduced from Fig. 11 of the published article under CC BY 4.0 (see title note).

Figure 11: Comparison between predicted and CFD spanwise performance curves for cases valid-A, valid-B, valid-C, and valid-D. Case D is the worst TNO prediction case selected from the validation set. Reproduced from Fig. 12 of the published article under CC BY 4.0 (see title note).

Model	Hyperparameter setting	Library
PR	degree=3	sklearn.linear_model. LinearRegression
ANN	depth=5; width=32; learning rate=0.01; iter_max=500	torch.nn.Module
SVR	kernel:linear; degree=3; C=1.0; epsilon=0.1; cache_size=200	sklearn.svm.SVR
XGB	objective:reg:squarederror; max_depth=6; n_estimators=160; subsample=0.8; reg_lambda=1e-5; learning rate=0.1	xgboost.XGBRegressor
GPR	kernel:scale-RBF; likelihood:gaussianLikelihood; variational distribution:multivariateNormal; learning rate=0.1; iter_max=500	gpytorch.models. ExactGP

Table 7: Conventional surrogate model hyperparameters and implementation libraries

Figure 12: The comparison between the accuracy of deep neural networks and conventional surrogate models

performanca	Algo	500	1000	1500	2000	2500
efficiency	MLP	1.064‰	1.091‰	1.042‰	1.031‰	1.318‰
	UNet	1.350‰	1.980‰	1.331‰	1.245‰	1.253‰
	deepONet	163.1‰	88.92‰	27.56‰	6.535‰	1.840‰
	FNO	1.213‰	1.188‰	1.057‰	0.938‰	0.810‰
	TNO	0.823‰	0.460‰	0.448‰	0.439‰	0.486‰
	POLY	1.086‰	1.079‰	1.048‰	1.070‰	1.069‰

Table 8: The predict accuracy of deep neural networks and conventional surrogate models with different training sample size

Figure 13: The influence area of each design variable on the blade suction surface.

Panel of Figure 14 — Figure 14: Sensitivity analysis of the efficiency span distribution and isentropic Mach number flow distribution. Reproduced from Fig. 14 of the published article under CC BY 4.0; the two original panels are arranged side by side (see title note).

performance parameter	search objective	reference value	optimal predict value	optimal CFD value	related error	related improvement
efficiency	max	0.854	0.871	0.871	-0.001%	1.95%
pressure loss	min	0.124	0.115	0.103	11.87%	17.24%
static entropy	min	44.83	41.44	38.94	6.42%	13.13%
pressure ratio	max	2.046	2.073	2.071	0.11%	1.21%
temperature ratio	max	1.265	1.270	1.271	-0.03%	0.44%

Table 9: The optimization and verify results of single-objective optimization tasks.

setting and performance	ref	opt-A	opt-B	opt-C
objective	-	$\max\eta_{is},\max\pi$	$\max\eta_{is},\min\pi$	$\max\eta_{is},\min\pi$
constrain	-	-	-	$\dot{m}\leq 20.265$
isentropic efficiency	0.854(-0.98‰)	0.871(-0.37‰)	0.869(1.66‰)	0.863(-1.76‰)
polytropic efficiency	0.872(-0.89‰)	0.883(-0.30‰)	0.882(1.48‰)	0.876(-1.58‰)
pressure loss	0.124(3.56‰)	0.116(7.265‰)	0.116(5.88‰)	0.119(14.7‰)
pressure ratio	2.046(-1.04‰)	2.069(-0.93‰)	2.045(-0.68‰)	2.04(-0.66‰)

Table 10: Performance metrics of reference and optimal designs

Panel of Figure 15 — Figure 15: The suction surface limit flow diagrams of the reference and optimal samples

Panel of Figure 16 — Figure 16: Optimal results of multi-objective optimization task-1

Panel of Figure 17 — Figure 17: Optimal results of multi-objective optimization task-2 and task-3

Figure 18: Grid-independence verification; the black box denotes the selected grid. Reproduced from Fig. A1 of the published article under CC BY 4.0 (see title note).

Network type	Output size
Input	28
(Linear+ BN1d+ GELU) $\times$ 5	256
Linear-7	4096 $\times$ 5
Forward pass size: 768.00MB Params size: 21.86MB

Table 11: The detailed network structure of MLP

Network type	Output size
Input	4096 $\times$ (28+2)
Conv2d+ ResBlock	4096 $\times$ 30
Conv2d $\times$ 2	4096 $\times$ 64
ResBlock + MaxPool2d + Conv2d $\times$ 2	64 $\times$ 32 $\times$ 32
ResBlock + MaxPool2d + Conv2d $\times$ 2	128 $\times$ 16 $\times$ 16
ResBlock+ MaxPool2d + Conv2d $\times$ 2	256 $\times$ 8 $\times$ 8

Table 12: The detailed network structure of UNet

Branch Network type	Output size
Input	28
(Linear+ FcnSingle) $\times$ 4	128
Linear	5
Trunk Network type	Output size
Input	4096 $\times$ 2
(Linear + GELU) $\times$ 10	4096 $\times$ 256

Table 13: The detailed network structure of deepONet

Network type	Output size
Input	4096 $\times$ (28+2)
Linear-1	4096 $\times$ 128
(Conv2d + Dropout + GELU + SpectralConv2d) $\times$ 4	128 $\times$ 72 $\times$ 72
Linear-2	4096 $\times$ 128
Linear-3	4096 $\times$ 5
Forward pass size: 558.91MB Params size: 1.66MB

Table 14: The detailed network structure of FNO

Figure 19: Predicted, CFD, and error maps of basic fields for cases valid-B and valid-C. Reproduced from the upper panels of Fig. A2 of the published article under CC BY 4.0 (see title note).

Figure 20: Predicted, CFD, and error maps of derived fields for cases valid-B and valid-C. Reproduced from the lower panels of Fig. A2 of the published article under CC BY 4.0 (see title note).

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Periodic Neural Mapping for Unsteady Rotor-Blade Pressure and Aeroelastic Load Prediction

Lionel Salesses, Joachim Dominique, Tariq Benamara, Théo Flament+1

Cenaero, Gosselies, Belgium · Safran Helicopter Engines, Bordes, France

Reduced-Order Modeling Neural Operators Computational Fluid Dynamics

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Accurate prediction of unsteady aerodynamic loads remains a major challenge in turbomachinery design. High-fidelity Computational Fluid Dynamics (CFD) simulations are expensive, while aeroelastic Quantities of Interest (QoI) depend sensitively on the temporal evolution of the pressure field. This work introduces periodic Fourier Neural Mapping (p-FNM), a neural-operator framework for predicting unsteady pressure distributions on turbine rotor blades simulated using the chorochronic numerical hypothesis. The architecture embeds temporal periodicity into the model and learns a continuous mapping from operating conditions and time to pressure fields. Unlike sequential latent-space approaches, p-FNM predicts pressure fields independently at any time, avoiding error accumulation while preserving temporal continuity. The model is evaluated on a database of unsteady rotor-blade simulations and compared with a reduced-order baseline based on a variational autoencoder and recurrent neural network, refered as the Temporal Prediction Model (TPM). Performance is assessed for pressure fields and Generalized Aerodynamic Forces (GAFs), the primary aeroelastic QoI. Across all training datasets, p-FNM consistently outperforms TPM. On the largest dataset, p-FNM achieves a pressure-field mean absolute percentage error of 0.46% and a GAF-magnitude prediction error of 4.42%, corresponding to improvements of 60.7% and 77.6%, respectively. The minimum weighted phase error reaches 0.060 rad, demonstrating accurate preservation of the temporal characteristics of the aerodynamic response. The results show that GAF prediction is more challenging than pressure-field prediction and that temporal coherence is critical for accurately predicting spectral aerodynamic quantities. These findings demonstrate the potential of periodic neural operators for reduced-order modeling and aeroelastic analysis in turbomachinery.

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