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REVIEW 3 major objections 4 minor 56 references

Fourier-enhanced reduced-order surrogate modeling for uncertainty quantification in electric machine design

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Compressing torque signals with a discrete Fourier transform and regressing the retained frequency components—best with Gaussian processes—yields the most accurate surrogate for a permanent magnet synchronous machine's torque under 20…

desk verdict Useful engineering comparison with a real flaw: the DFT basis is selected on the full dataset, so the reported validation accuracy does not actually measure generalization to unseen designs. read the letter →

arxiv 2412.06485 v2 pith:X2JGRMYD submitted 2024-12-09 cs.CE

Aylar Partovizadeh , Sebastian Schöps , Dimitrios Loukrezis This is my paper · ORCID
classification cs.CE
keywords discreteFouriertransformdimensionreductionsurrogatemodelingGaussianprocessregressionpermanentmagnetsynchronousmachineuncertaintyquantificationpolynomialchaosexpansionelectricdesign
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a cheap surrogate can replace an expensive isogeometric simulation when predicting the full torque waveform of a permanent magnet synchronous machine whose 20 geometric design parameters vary randomly. Its recipe is to compress each 120-sample torque signal with a discrete Fourier transform, keep the 11 frequency components with the largest averaged power contributions across 2000 training simulations, train a regression model from the design parameters to those components, and then invert the transform to recover the torque signal. Comparing polynomial chaos, feedforward neural networks, and Gaussian processes as regressors, the authors find that the Fourier-plus-Gaussian-process combination performs best, reaching a signal-averaged mean absolute percentage error of 0.0029 with 1800 training samples. If correct, this makes Monte Carlo uncertainty quantification of torque statistics several times cheaper, with total savings of 5.5 to 16.7 times when offline data generation and training costs are included.

What carries the argument

The load-bearing object is the reduced Fourier representation. Each 120-sample torque signal $\tau^{(m)}$ is transformed by the discrete Fourier transform into complex coefficients $c_k^{(m)}$, the per-frequency power contributions $\eta_k^{(m)} = |c_k^{(m)}|^2$ are averaged over the $M=2000$ training signals, and the $R=11$ frequencies with the largest averaged contributions form the index set $I$ that defines the reduced quantity of interest $r\in\mathbb{C}^R$. A response surface $S:p\mapsto r$, built with polynomial chaos, a neural network, or a Gaussian process, is trained on this reduced target, and prediction is $R^{-1}(S(p^*))$, the inverse DFT using only the retained modes. The fixed index set is what decouples the truncation error from the regression error and keeps the training target low-dimensional while preserving the signal's periodic content.

What would settle it

Take a design parameter combination outside the uniform training ranges in Table 1 (or a Gaussian sample near a range boundary), compute the full 120-sample torque with the high-fidelity IGA model, and compare it with the inverse DFT reconstructed from the fixed 11-component index set. If the worst-case reconstruction MAE grows well above the $9.0\times10^{-5}$ reported for the training distribution—for instance because a formerly negligible harmonic becomes dominant—the fixed-basis assumption fails and the surrogate's accuracy ceiling is set by the truncation, not the regressor.

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Extended reading notes

Core claim

The central claim is that periodicity makes the discrete Fourier transform a better dimension-reduction basis than principal component analysis for surrogate modeling of electric-machine torque. The authors fix a reduced basis of $R=11$ frequency components selected from the averaged power spectrum of the training set, fit response surfaces from design parameters to the complex Fourier coefficients, and reconstruct torque by inverse DFT. In their numerical comparison, DFT-based reduction outperforms direct approximation of the 120-dimensional torque signal for every regressor and training size tested, and edges out PCA for the best regressors. The best surrogate, DFT reduction with a Gaussian process response surface, reaches a signal-averaged MAPE of $0.0029$ at $M_t=1800$ and, in Monte Carlo studies, reproduces mean torque estimates to within a few ten-thousandths of the reference while cutting the cost of a $10^4$-sample study dramatically.

Load-bearing premise

The assumption that the 11 frequency components chosen from the average power spectrum of 2000 uniformly sampled training designs remain the right reduced basis for every unseen design and for the narrower Gaussian sampling distribution; if geometric variations shift the torque's spectral content into omitted frequencies, the fixed truncation adds reconstruction error the surrogate cannot correct.

Editorial extensions

If this is right

  • With 1,800 training signals, the DFT-plus-GP surrogate predicts full torque waveforms on unseen designs with a signal-averaged MAPE of 0.0029, slightly better than the best PCA-based GP (0.0030) and far below direct no-reduction surrogates.
  • Monte Carlo UQ with $10^4$ samples costs about 10 seconds of online surrogate time instead of roughly 833 CPU-hours for the high-fidelity model; including offline data generation and training, total savings are 5.5 to 16.7 times depending on training set size.
  • For uniform parameter distributions, surrogate-based estimates of the torque mean stay within a signal-averaged APE of about $4\times10^{-5}$ for the best GP at $M_t=1800$, with standard-deviation estimates within about 0.6%.
  • Training on uniform data and then sampling under a Gaussian distribution degrades statistics accuracy by roughly an order of magnitude, but the Gaussian-process surrogate still shows the smallest maximum errors and the least fluctuation.
  • Omitting dimension reduction altogether severely hurts all regressors, most visibly the neural network, so the reduction step itself carries much of the accuracy gain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's experiments, the averaged-power-spectrum recipe should transfer to other periodic electric-machine outputs—cogging torque, flux linkage, force ripple—where the retained harmonic set can be chosen automatically instead of from expert harmonic analysis.
  • A targeted robustness test would train on uniform data and then re-fit only the response surface on Gaussian-sampled data; the paper's results suggest most of the Gaussian-case error comes from distribution shift in the regressor rather than from the 11-mode DFT truncation.
  • An adaptive variant that re-selects the frequency index set per parameter region would be a natural extension if spectra shift with geometry; the fixed 11-mode basis is the clear point to revisit first.
  • For manufacturing-tolerance studies with tight distributions around nominal values, re-selecting the reduced basis on the target distribution is a low-cost safeguard, since the truncation verified on wide uniform samples may keep irrelevant modes and miss narrow-range spectral features.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a data-driven surrogate modeling framework for predicting the electromagnetic torque of a permanent magnet synchronous machine under geometric design variations. The framework combines discrete Fourier transform (DFT) based dimension reduction of torque signals with three response surface models (polynomial chaos expansion, feedforward neural network, Gaussian process). The authors report that the DFT-plus-GP combination yields the best surrogate accuracy, outperforming direct torque approximation and PCA-based reduction, and that the surrogates enable Monte Carlo uncertainty quantification at 5.5x–16.7x lower computational cost including offline expenses. The central claim is that the framework produces accurate predictions for previously unseen design configurations and reliable torque statistics.

Significance. If the reported results are robust, the framework is a useful contribution to surrogate-based electric machine design, with a physically motivated dimension reduction step and a systematic comparison across three regression models and three reduction schemes. The paper also provides a clear computational cost analysis and explicitly tests both uniform and Gaussian parameter distributions in the UQ studies. The main methodological strength is the use of DFT to exploit torque periodicity while retaining physical interpretability of the reduced representation. However, the validation protocol has a data-leakage issue that directly affects the reported generalization errors, and the advantage of DFT over PCA is numerically small, so the central claim requires a corrected validation before the results can be taken at face value.

major comments (3)
  1. [Section 4.1 and Section 4.2] The DFT basis is selected using the full dataset of M=2000 torque signals, including the Mv=200 validation samples later used in Section 4.2. Specifically, Eq. (13) averages power-spectrum contributions over all M signals to determine the index set I and the truncation R=11, and the reconstruction MAE in Eq. (26) is also computed on the same M=2000 signals. Consequently, the MAPE and SDAPE values in Table 3 and Figures 5–6 are computed on validation data that influenced the reduced basis, so they do not measure generalization to genuinely unseen design configurations as claimed in Section 1 and the Abstract. This is load-bearing because the DFT-vs-PCA gap is small (0.0029 vs 0.0030 at Mt=1800), and the reported advantage could be an artifact of leakage. The authors should either reselect the basis and R using only the training portion of the data, or use a nested cross-validation scheme, and report the resulting errors. They should also state explicitly whether the PCA components are fitted on all 2000 samples or only on the training set.
  2. [Section 4.2 and Table 3] The FNN is trained with stochastic gradient descent, but no repeated-seed or confidence-interval information is reported for the MAPE/SDAPE values. Since the FNN is the worst-performing RSM and the differences among PCE and GP are sometimes small, it is unclear whether the reported ordering of RSMs is stable. The authors should report means and standard deviations over multiple training runs with different random seeds, or at least indicate the number of runs used.
  3. [Section 4.3.2] In the Gaussian-distribution UQ study, the surrogate models are trained on uniformly distributed parameters but sampled under a Gaussian distribution concentrated around the nominal values. The authors acknowledge this distributional shift, but the reduced basis I was also selected from the uniform dataset. Since the Gaussian sampling region is a small subset of the training range, the fixed DFT basis may not be appropriate there. The paper should either test the reconstruction accuracy of the fixed basis within the Gaussian sampling region or discuss why the basis is expected to remain adequate; otherwise the UQ statistics errors in Table 5 may partly reflect basis mismatch rather than RSM accuracy.
minor comments (4)
  1. [Section 2.1] In the paragraph following Eq. (3), 'Not that' should be 'Note that'.
  2. [Section 4.1 and Figure 4] The caption of Figure 4 refers to 'worst-case signal reconstruction' but the worst-case signal is not defined; please clarify whether it is the signal with maximum reconstruction error.
  3. [Section 4.2] The sentence 'In the former case, 21 principal components are kept, resulting in an MAE equal to 2.8·10−5' does not state the dataset on which this MAE is computed; please specify whether it is the same M=2000 dataset used for the DFT reconstruction study.
  4. [Section 3.2.3 and Eq. (23)] The notation k(p*,P) and k(P,p*) is introduced after Eq. (23) but is used before the definitions; please define these terms in the text preceding the equation for clarity.

Circularity Check

1 steps flagged · score 4.0 of 10

Validation leakage in DFT basis selection undermines the 'previously unseen' MAPE, but the RSM/UQ derivation is otherwise self-contained.

  1. fitted input called prediction [Section 4.1, Eq. (13), and Section 4.2, dataset partition]
    "To that end, we use a sample of M = 2000 torque signals ... ηk = 1/M Σ_{m=1}^{M} |c^{(m)}_k|^2 ... Based on these observations and numerical results, in the following, R = 11 frequency components will be retained after the DFT. ... For training and validation, an initial dataset D = {p^{(m)}, τ^{(m)} = τ(p^{(m)})}_{m=1}^{M}, M = 2000, is partitioned into a training dataset Dt and a validation dataset Dv, where Dt ∩ Dv = ∅."

    The DFT reduced basis (index set I and truncation R=11) is selected from the averaged power spectrum over all M=2000 torque signals (Eq. 13) and from the reconstruction MAE in Eq. (26) over the same M=2000 signals. Section 4.2 then partitions this exact M=2000 dataset into training Dt and validation Dv. Hence the 200 validation designs used to compute the MAPE values in Table 3 and Figure 5 already contributed to the choice of the reduced basis. The abstract calls these 'previously unseen design configurations,' but at the level of dimension reduction they are seen by construction.

full rationale

The one substantive circularity is in the validation protocol: the DFT-based dimension reduction (I and R=11) is fitted using the full M=2000 dataset before that same dataset is split into training and validation in Section 4.2. Therefore the validation MAPE does not measure true generalization of the complete DFT+RSM pipeline to previously unseen designs. I found no other circular steps. The regression models (PCE, FNN, GP) are trained only on the training subset; the UQ statistics in Section 4.3 are compared against fresh high-fidelity Monte Carlo reference statistics; and the self-citations ([21], [30], [33]-[35]) serve as background/model sources rather than load-bearing justifications of the framework's claims. The central 'DFT+GP best-in-class' conclusion still has independent empirical content, because it is a comparative study under the same protocol and because the surrogate-based UQ estimates are checked against the original high-fidelity model. The score of 4 reflects the partial circularity in the reported generalization error, not a collapse of the whole derivation.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the fidelity of the IGA model and on the stability of the torque signal's frequency content across the design space. No new physical entities are introduced. The main fitted quantities are the reduction size R, the PCA component count, and the regression hyperparameters.

free parameters (5)
  • R: number of retained DFT components = 11
    Selected in Section 4.1 based on reconstruction MAE on the M=2000 dataset; controls the dimension reduction and surrogate accuracy.
  • Number of PCA components = 21
    Chosen in Section 4.2 to match the real dimension 2R-1 of the DFT-reduced representation; affects the PCA comparison.
  • FNN architecture (layers and neurons) = 4 layers, 45/60/80/25 neurons
    Determined by incremental increases until no significant validation error improvement, Section 3.2.2.
  • GP covariance hyperparameters = Optimized via CMA-ES
    Fitted by maximum likelihood on training data, Equation (25), Section 3.2.3.
  • PCE polynomial basis truncation = Adaptive sparse via LAR
    Selected adaptively by the UQLab implementation, Section 3.2.1.
assumptions (4)
  • domain assumption The 2D magnetostatic model with quarter symmetry, harmonic mortaring, and neglected eddy/end effects adequately represents torque.
    Taken from prior work [21]; all numerical studies inherit this modeling assumption. Section 2.2.
  • domain assumption Torque signal is periodic with known 30-degree period and the DFT index set I of R=11 components selected from training data remains adequate for all unseen designs and sampling distributions.
    Central to the fixed dimension reduction; if spectral content shifts, truncation error increases. Sections 3.1, 4.1.
  • domain assumption Training design parameters are uniformly distributed and independent; surrogate trained on this distribution is then evaluated on Gaussian-distributed parameters.
    Distribution shift is acknowledged and shown to degrade accuracy. Section 4.3.2.
  • standard math Standard DFT/IDFT and probability theory results hold.
    Equations (10)-(15) rely on standard Fourier analysis and orthogonal polynomial theory.

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Pith. "Pith review of Fourier-enhanced reduced-order surrogate modeling for uncertainty quantification in electric machine design." pith.science (2026). https://pith.science/paper/X2JGRMYD

@misc{pith2026241206485,
  author       = {Pith},
  title        = {Pith review of: Fourier-enhanced reduced-order surrogate modeling for uncertainty quantification in electric machine design},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2JGRMYD}},
  note         = {Machine review of arXiv:2412.06485}
}
read the original abstract

This work proposes a data-driven surrogate modeling framework for cost-effectively inferring the torque of a permanent magnet synchronous machine under geometric design variations. The framework is separated into a reduced-order modeling and an inference part. Given a dataset of torque signals, each corresponding to a different set of design parameters, torque dimension is first reduced by post-processing a discrete Fourier transform and keeping a reduced number of frequency components. This allows to take advantage of torque periodicity and preserve physical information contained in the frequency components. Next, a response surface model is computed by means of machine learning regression, which maps the design parameters to the reduced frequency components. The response surface models of choice are polynomial chaos expansions, feedforward neural networks, and Gaussian processes. Torque inference is performed by evaluating the response surface model for new design parameters and then inverting the dimension reduction. Numerical results show that the resulting surrogate models lead to sufficiently accurate torque predictions for previously unseen design configurations. The framework is found to be significantly advantageous compared to approximating the original (not reduced) torque signal directly, as well as slightly advantageous compared to using principal component analysis for dimension reduction. The combination of discrete Fourier transform-based dimension reduction with Gaussian process-based response surfaces yields the best-in-class surrogate model for this use case. The surrogate models replace the original, high-fidelity model in Monte Carlo-based uncertainty quantification studies, where they provide accurate torque statistics estimates at significantly reduced computational cost.

Figures

Figures reproduced from arXiv: 2412.06485 by the authors.

Figure 1
Figure 1. One-fourth of the PMSM’s geometry in two dimensions, including material distribution, bound￾ary conditions, and geometric design parameters (see [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Torque signals for different geometric design configurations of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Schematic illustration of the surrogate modeling framework, separated into reduced-order [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Reconstruction error versus dimension reduction and worst-case torque signal reconstructions [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Mean absolute percentage error (MAPE) for surrogates computed with different combinations of training dataset size, RSM, and dimension reduction approach. in-class surrogate model. These observations are further supported by the results presented in [PITH_FULL_IMAGE:f…
Figure 6
Figure 6. Figure 6: Standard deviation of absolute percentage error (SDAPE) for surrogates computed with differ￾ent combinations of training dataset size, RSM, and dimension reduction approach. Last but not least, it is worth noting that surrogate modeling accuracy and robustness clearly …
Figure 7
Figure 7. Figure 7: Worst-case surrogate-based torque signal predictions for training dataset size [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Torque mean and standard deviation estimates for uniform parameter distributions, obtained [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: APE of torque mean and standard deviation estimates for uniform parameter distributions, obtained with framework-based surrogates computed with different combinations of training dataset size and RSM. 4.3.2 Gaussian parameter distributions In the second UQ study, we as…
Figure 10
Figure 10. Figure 10: Torque mean and standard deviation estimates for Gaussian parameter distributions, obtained [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: APE of torque mean and standard deviation estimates for Gaussian parameter distributions, obtained with framework-based surrogates computed with different combinations of training dataset size and RSM. presented in [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Worst-case surrogate-based torque signal signal predictions for training dataset size [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Worst-case surrogate-based torque signal predictions for training dataset size [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.