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

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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

Figure 1: Mid-span ( h/H=0.5h/H=0.5 ) schlieren-like density gradient and static pressure field on the reference blade, showing the correlation between flow structures and aerodynamic loading
Table 1: Number of simulations for the different and independent datasets.
Figure 2: FNO layer architecture
Figure 3: p-FNM architecture (a) and V2F block (b)
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 ±1.96σ\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.
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 ±1.96σ\pm 1.96\sigma confidence interval computed over 30 independent training runs.
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 ±1.96σ\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 ±5%\pm 5\% and ±10%\pm 10\% , respectively. The markers represent the mean predicted GAF values, and the gray error bars denote the associated prediction uncertainty quantified by the ±1.96σ\pm 1.96\sigma confidence interval. Each grid cell corresponds to 0.0005 GAF units along both axes.
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 [−π,π][-\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 ±1.96σ\pm 1.96\sigma confidence interval computed over all independent training runs.
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 ±1.96σ\pm 1.96\sigma confidence interval computed over all independent training runs.
Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

ModelParameters count
VAE1,083,649
TPM (RNN module only)75,104
TPM (full architecture)1,158,753
p-FNM1,156,094
GAF-MLP1,169,427
Projected-force-MLP1,199,425
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.
DatasetMetricVAEVAE with SOAPTPMTPM-SOAP
SmallPressure MAE [kPa]0.97 ±\pm 0.080.87 ±\pm 0.032.05 ±\pm 0.341.80 ±\pm 0.31
Pressure MAPE [%]1.59 ±\pm 0.141.42 ±\pm 0.0473.35 ±\pm 0.542.91 ±\pm 0.48
Pressure MME [kPa]18.74 ±\pm 1.6316.98 ±\pm 1.1028.53 ±\pm 3.5825.30 ±\pm 3.04
Pressure R2R^{2}0.9904 ±\pm 0.00140.9914 ±\pm 0.00130.9501 ±\pm 0.0180.9610 ±\pm 0.015
GAF norm MAPEρ\mathrm{MAPE}_{\rho} [%]27.29 ±\pm 11.2825.56 ±\pm 6.5967.11 ±\pm 32.2438.52 ±\pm 13.75
GAF norm Rρ2R^{2}_{\rho}0.70 ±\pm 0.160.71 ±\pm 0.0890.018 ±\pm 0.430.45 ±\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.
DatasetVAEVAE with SOAPTPM aloneTPM alone (SOAP variant)TPM totalTPM-SOAP total
Small0.16 ±\pm 0.080.20 ±\pm 0.080.02 ±\pm 0.0030.03 ±\pm 0.010.18 ±\pm 0.080.23 ±\pm 0.09
Medium0.29 ±\pm 0.120.32 ±\pm 0.110.04 ±\pm 0.010.05 ±\pm 0.020.33 ±\pm 0.130.37 ±\pm 0.13
Large0.46 ±\pm 0.180.64 ±\pm 0.280.11 ±\pm 0.030.14 ±\pm 0.070.57 ±\pm 0.210.78 ±\pm 0.35
Extra Large0.83 ±\pm 0.251.38 ±\pm 0.560.29 ±\pm 0.030.27 ±\pm 0.11.12 ±\pm 0.281.65 ±\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 blockLinear(6, 1440) -> Truncated Fourier modes = 12x12
Lift operator P\mathcal{P}LinearChannelWise(6, 22)
FNO Layers count4
FNO Layer W\mathcal{W}Conv2D(channels_in=22, channels_out=22, kernel_size=3, padding=1)
FNO Layer SpectralConvSpectralConv2D(channels_in=22, channels_out=22, modes_w=12, modes_h=12)
FNO Layer ActivationReLU
Projection operator Q\mathcal{Q}LinearChannelWise(22, 128) -> LeakyReLU -> Linear(128, 1)
Appendix
Table 7: p-FNM model hyperparameters
DatasetMetricp-FNM
SmallPressure MAE [kPa]1.21 ±\pm 0.095
Pressure MAPE [%]1.98 ±\pm 0.16
Pressure MME [kPa]18.37 ±\pm 1.80
Pressure R2R^{2}0.9827 ±\pm 0.0026
GAF norm MAPEρ\mathrm{MAPE}_{\rho} [%]32.13 ±\pm 7.30
GAF norm Rρ2R^{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.
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.
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.
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.
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