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

Lionel Salesses; Joachim Dominique; Tariq Benamara; Théo Flament; Franck Mastrippolito

physics.flu-dynSep 18, 2026

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

Authors: Lionel Salesses, Joachim Dominique, Tariq Benamara, Théo Flament, Franck Mastrippolito

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

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Mid-span ( $h/H=0.5$ ) schlieren-like density gradient and static pressure field on the reference blade, showing the correlation between flow structures and aerodynamic loading

Panel of Table 1 — Table 1: Number of simulations for the different and independent datasets.

Figure 3: p-FNM architecture (a) and V2F block (b)

Panel of Figure 4 — Figure 4: Comparison of the TPM, TPM-SOAP and p-FNM models with respect to the pressure prediction metrics across the different training datasets. The metrics are evaluated on the test dataset. The black error bars represent the metric uncertainty quantified by the $\pm 1.96\sigma$ confidence interval computed over 30 independent training runs.

Table 2: Comparison between predicted and reference projected-force signals for the VAE, TPM, and p-FNM models across the different training datasets for two representative test-cases from the test dataset. The black curve denotes the reference solution, whereas the orange, red, and green curves correspond to the VAE, TPM, and p-FNM predictions, respectively. For visual reference, each grid cell corresponds to 0.0005 units of projected force along the vertical axis.

Panel of Figure 5 — Figure 5: Comparison of the TPM, and p-FNM models with respect to the temporal spectrum of the projected-force signal evaluated on the test dataset. All the Fourier harmonics of the blade-passing frequency used in the chorochronic boundary conditions are displayed.

Figure 6: Weighted Mean Absolute Percentage Error (wMAPE) associated with the prediction of the first Fourier mode of the projected-force for the TPM and p-FNM models across the different training datasets. The wMAPE is evaluated on the test dataset. The black error bars represent the metric uncertainty quantified by the

\pm 1.96\sigma

confidence interval computed over 30 independent training runs.

Panel of Figure 7 — Figure 7: Comparison of the TPM, TPM-SOAP and p-FNM models with respect to the GAF prediction metrics across the different training datasets. The metrics are evaluated on the test dataset. The black error bars represent the metric uncertainty quantified by the $\pm 1.96\sigma$ confidence interval computed over 30 independent training runs.

Table 3: Comparison between predicted and reference GAF norm values for the TPM and p-FNM models across the different training datasets. The models are evaluated on the test dataset. The black dashed line indicates perfect agreement, while the green and red dashed lines correspond to relative errors of

\pm 5\%

and

\pm 10\%

, respectively. The markers represent the mean predicted GAF values, and the gray error bars denote the associated prediction uncertainty quantified by the

\pm 1.96\sigma

confidence interval. Each grid cell corresponds to 0.0005 GAF units along both axes.

Panel of Figure 8 — Figure 8: Comparison of the GAF phase prediction error distributions (in radians) for the TPM and p-FNM models across the different training datasets. The models are evaluated on the test dataset. The phase error is defined within the interval $[-\pi,\pi]$ . Optimal performance corresponds to error distributions centered around zero with minimal dispersion.

Figure 9: Comparison of the TPM, TPM-SOAP, and p-FNM models in terms of training time (GPU

\cdot

h) across the different training datasets. The black error bars indicate the training-time uncertainty quantified by the

\pm 1.96\sigma

confidence interval computed over all independent training runs.

Panel of Figure 10 — Figure 10: Comparison of the p-FNM, GAF-MLP and projected-force-MLP models on GAF metrics across the different training datasets. The metrics are evaluated on the test dataset. The black error bars indicate the training-time uncertainty quantified by the $\pm 1.96\sigma$ confidence interval computed over all independent training runs.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Parameters count
VAE	1,083,649
TPM (RNN module only)	75,104
TPM (full architecture)	1,158,753
p-FNM	1,156,094
GAF-MLP	1,169,427
Projected-force-MLP	1,199,425

Appendix

Table 4: Number of trainable parameters for the deep learning architectures considered in this study. The TPM parameter count is reported both for the recurrent prediction module alone and for the complete model, including the VAE component.

Dataset	Metric	VAE	VAE with SOAP	TPM	TPM-SOAP
Small	Pressure MAE [kPa]	0.97 $\pm$ 0.08	0.87 $\pm$ 0.03	2.05 $\pm$ 0.34	1.80 $\pm$ 0.31
	Pressure MAPE [%]	1.59 $\pm$ 0.14	1.42 $\pm$ 0.047	3.35 $\pm$ 0.54	2.91 $\pm$ 0.48
	Pressure MME [kPa]	18.74 $\pm$ 1.63	16.98 $\pm$ 1.10	28.53 $\pm$ 3.58	25.30 $\pm$ 3.04
	Pressure $R^{2}$	0.9904 $\pm$ 0.0014	0.9914 $\pm$ 0.0013	0.9501 $\pm$ 0.018	0.9610 $\pm$ 0.015
	GAF norm $\mathrm{MAPE}_{\rho}$ [%]	27.29 $\pm$ 11.28	25.56 $\pm$ 6.59	67.11 $\pm$ 32.24	38.52 $\pm$ 13.75
	GAF norm $R^{2}_{\rho}$	0.70 $\pm$ 0.16	0.71 $\pm$ 0.089	0.018 $\pm$ 0.43	0.45 $\pm$ 0.31

Appendix

Table 5: Comparison of the VAE and TPM models [ 12 ] with their variants trained using the SOAP optimizer. Metrics are reported as the mean

\pm

1.96 std for the test dataset across 30 training runs.

Dataset	VAE	VAE with SOAP	TPM alone	TPM alone (SOAP variant)	TPM total	TPM-SOAP total
Small	0.16 $\pm$ 0.08	0.20 $\pm$ 0.08	0.02 $\pm$ 0.003	0.03 $\pm$ 0.01	0.18 $\pm$ 0.08	0.23 $\pm$ 0.09
Medium	0.29 $\pm$ 0.12	0.32 $\pm$ 0.11	0.04 $\pm$ 0.01	0.05 $\pm$ 0.02	0.33 $\pm$ 0.13	0.37 $\pm$ 0.13
Large	0.46 $\pm$ 0.18	0.64 $\pm$ 0.28	0.11 $\pm$ 0.03	0.14 $\pm$ 0.07	0.57 $\pm$ 0.21	0.78 $\pm$ 0.35
Extra Large	0.83 $\pm$ 0.25	1.38 $\pm$ 0.56	0.29 $\pm$ 0.03	0.27 $\pm$ 0.1	1.12 $\pm$ 0.28	1.65 $\pm$ 0.66

Appendix

Table 6: Comparison of models training time [GPU.h] for the VAE and TPM models [ 12 ] with their variants trained using the SOAP optimizer. Metrics are reported as the mean

\pm

1.96 std for the test set across 30 training runs.

V2F block	Linear(6, 1440) -> Truncated Fourier modes = 12x12
Lift operator $\mathcal{P}$	LinearChannelWise(6, 22)
FNO Layers count	4
FNO Layer $\mathcal{W}$	Conv2D(channels_in=22, channels_out=22, kernel_size=3, padding=1)
FNO Layer SpectralConv	SpectralConv2D(channels_in=22, channels_out=22, modes_w=12, modes_h=12)
FNO Layer Activation	ReLU
Projection operator $\mathcal{Q}$	LinearChannelWise(22, 128) -> LeakyReLU -> Linear(128, 1)

Appendix

Table 7: p-FNM model hyperparameters

Dataset	Metric	p-FNM
Small	Pressure MAE [kPa]	1.21 $\pm$ 0.095
	Pressure MAPE [%]	1.98 $\pm$ 0.16
	Pressure MME [kPa]	18.37 $\pm$ 1.80
	Pressure $R^{2}$	0.9827 $\pm$ 0.0026
	GAF norm $\mathrm{MAPE}_{\rho}$ [%]	32.13 $\pm$ 7.30
	GAF norm $R^{2}_{\rho}$	0.57 $\pm$ 0.21

Appendix

Table 8: Detailed metrics for the p-FNM model across the training datasets. Metrics are reported as the mean

\pm

1.96 std for the test dataset across 30 training runs.

Panel of Table 9 — Table 9: Example of predicted and reference pressure fields for a test-dataset simulation at a fixed time step. The red vertical line corresponds to the leading edge of the blade.

Panel of Table 10 — Table 10: Example of predicted and reference pressure fields for a test-dataset simulation at a fixed time step. The red vertical line corresponds to the leading edge of the blade.

Panel of Table 11 — Table 11: t-SNE projection of latent means and log-variances obtained by encoding all time steps of different test-dataset simulations using a VAE trained on the large dataset.

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

See all 88 papers

A Residual Learning Approach for Unsteady Aerodynamic Load Prediction
Aug 18, 2026Divya Sanghi, Carlos E. S. CesnikResidual Learning
University of Michigan, Ann Arbor, MI, 48109
A panoramic aerodynamic performance prediction method for turbomachinery cascades using transformer-enhanced neural operator
Sep 13, 2026Qineng Wang, Zhendong Guo, Liming Song +1Transformer Neural Operator PDE Surrogate Modeling
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
VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows
Sep 1, 2026Xingxin Yang, Zhan Zhang, Yichen Li +1Transformer Neural Operator PDE Operator Learning
aDepartment of Engineering, King’s College London, London, WC2R 2LS, UK

Aug 18, 2026J/K move · Enter open · S save

A Residual Learning Approach for Unsteady Aerodynamic Load Prediction

Divya Sanghi, Carlos E. S. Cesnik

University of Michigan, Ann Arbor, MI, 48109

Residual Learning

Open paper AskPDF arXiv

This paper investigates the feasibility of using residual learning to improve unsteady aerodynamic load prediction for aeroelastic applications. The machine learning technique selected for the study is the long short-term memory (LSTM) neural network, which is used for its suitability for sequential data with aerodynamic memory effects. The approach is investigated for the NLR 7301 airfoil benchmark using high-fidelity CFD lift data for prescribed pitch and plunge motions in the transonic flow regime in the presence of shock motion. An analytical unsteady aerodynamic model based on the Wagner function is used as a physics-based baseline, and the neural network is trained to learn the difference between the CFD lift coefficient and the Wagner prediction. The residual model is compared with a direct neural-network model trained to predict the CFD lift coefficient. The comparison includes feature and normalization studies, external benchmark cases, and leave-one-out and leave-family-out generalization tests across a range of sinusoidal and non-sinusoidal motions. The residual model performs best when its inputs align with the Wagner formulation variables, generally giving lower error and more consistent performance across training runs, though the direct model remains more accurate for some high-frequency cases. The residual model also generalizes better in the leave-one-out and leave-family-out tests, with a smaller increase in error than the direct model when entire motion families are withheld from training. Overall, the results indicate that residual learning shows promise as a modular approach for augmenting classical low-order aerodynamic theories, especially when the physics baseline removes a structured part of the aerodynamic response and leaves a lower-variance correction for the neural network to learn.

Abstract

Reduced-Order Modeling

Explore similar work

A Residual Learning Approach for Unsteady Aerodynamic Load Prediction

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

VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows