Predictive Modeling and Uncertainty Quantification of Fatigue Life in Metal Alloys using Machine Learning

Aleksandar Vakanski; Deekshith Basvoju; Indrajit Charit; Jiang Chang; Min Xian

REVIEW 4 major objections 4 minor 55 references

Predictive Modeling and Uncertainty Quantification of Fatigue Life in Metal Alloys using Machine Learning

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Augmenting machine learning with a Basquin-estimated fatigue-life feature and a boundary-enforcing physics-informed loss yields more accurate and better-calibrated predictions on all four alloy datasets tested.

desk verdict Useful UQ benchmark with a plausible PIML idea, but the loss is reversed, Table 8 duplicates Table 4, and the Basquin feature may leak the target, so the central claim is unproven. read the letter →

arxiv 2501.15057 v1 pith:A3PBN62P submitted 2025-01-25 cs.LG cond-mat.mtrl-sci

Jiang Chang , Deekshith Basvoju , Aleksandar Vakanski , Indrajit Charit , Min Xian This is my paper · ORCID

classification cs.LGcond-mat.mtrl-sci

keywords fatiguelifepredictionphysics-informedmachinelearninguncertaintyquantificationBayesianneuralnetworksGaussianprocessregressionmetalalloysBasquinmodelboundaryconstraints

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

Fatigue life of metal alloys is expensive to measure, and purely data-driven models often give point estimates without trustworthy confidence intervals. The paper proposes a physics-informed machine learning approach that adds a fatigue-life estimate from the Basquin stress-life relation as an extra input feature and trains neural networks with a loss that penalizes predictions outside the physically plausible range of zero to ten million cycles. Across four experimental datasets — a titanium alloy set and three carbon steel sets from uniaxial, bending, and torsion fatigue tests — the physics-informed versions improve accuracy and uncertainty metrics relative to the same models without physics. The paper reports that single-point metric improvements are on the order of ten to twenty percent, with the strongest overall point-and-interval performance from Markov chain Monte Carlo Bayesian neural networks and Gaussian process regression.

What carries the argument

Two ingredients carry the argument. The first is the Basquin stress-life relation, a power law $\sigma = c N_f^m$ relating stress amplitude to cycles to failure; fitted to each dataset, it supplies a physics-informed fatigue-life feature that is concatenated with the measured alloy, processing, and test-condition features. The second is a physics-informed loss function for the neural networks, $\mathcal{L} = \frac{1}{N}\sum_i [(y_i - y_i^*)^2 + \lambda_1 \mathrm{ReLU}(y_i^*) + \lambda_2 \mathrm{ReLU}(10{,}000{,}000 - y_i^*)]$, which penalizes negative predicted lives and predictions beyond the ten-million-cycle endurance-limit boundary. Together they embed the known monotone stress-life trend and its physical bounds into models that would otherwise fit only the measured samples.

What would settle it

Refit the Basquin constants separately inside each of the five cross-validation training folds, recompute the physics-informed feature for the test fold only, and rerun all reported models; if the roughly ten-to-twenty-percent improvements in $R^2$, RMSE, and coverage shrink or disappear, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that combining physics-based feature engineering with physics-informed regularization makes data-driven fatigue-life prediction both more accurate and more reliable in its uncertainty estimates. In the proposed framework, the Basquin relation $\sigma = c N_f^m$ is fitted to experimental stress-life data to produce a physics-informed estimate of fatigue life, and this estimate is appended to the measured input features. For neural-network models, the training loss adds ReLU penalty terms that force predicted fatigue life to stay non-negative and below $10{,}000{,}000$ cycles, the conventional endurance-limit boundary. Validated on four fatigue-test datasets, the method consistently improves $R^2$, RMSE, MAE, coverage, and a composite uncertainty metric compared with the identical models trained without the physics-informed feature and loss; the largest gains appear where the data-only baseline is weakest, such as the uniaxial carbon steel dataset.

Load-bearing premise

The reported improvements assume that the Basquin material constants are fitted in a way that does not let test-fold fatigue lives leak into the training features; if the constants are fit on the full dataset, the physics-informed feature carries information from the held-out samples and the gains would be inflated by leakage.

Editorial extensions

If this is right

On all four datasets, every evaluated model improves when the physics-informed feature and loss are added, with single-point metrics such as $R^2$, RMSE, and MAE improving by roughly ten to twenty percent.
The boundary-aware loss keeps predicted fatigue lives within the physically meaningful interval from zero to ten million cycles, eliminating predictions that are negative or beyond the endurance-limit cutoff.
Markov chain Monte Carlo Bayesian neural networks and Gaussian process regression produce the most reliable point estimates and uncertainty intervals, while deep ensembles and MC dropout tend to under-cover the true values.
The resulting point predictions plus confidence intervals can be used to plan fatigue tests for new alloys, potentially reducing the number of tests and cycles needed for material qualification.

Reading between the lines

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

If the Basquin feature is fitted strictly inside each training fold, the framework should transfer to any property governed by a simple power law, such as creep or tensile strength; the paper does not test this transfer explicitly.
The largest gains appear exactly where the data-only baseline is weakest, suggesting the physics feature acts most as a regularizer when data are sparse or noisy; one could test this by applying the method to deliberately downsampled versions of the four datasets.
A model with a randomly permuted or shuffled physics feature would separate the feature's information content from its mere presence, and would clarify how much of the gain is due to the physics itself versus added model capacity.
The upper-bound penalty at ten million cycles may mask run-out behavior; the paper does not report how predictions near the boundary behave, so it would be informative to evaluate performance on specimens whose measured lives exceed the cutoff.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper compares eight ML models for fatigue life prediction with a physics-informed feature (Basquin-estimated life) and a boundary loss, on four datasets. That's a useful benchmark, and the idea of adding a classical S-N curve estimate as an input is a reasonable thing to try. The code is available, which is good.

What the paper does well: it runs a broad, systematic comparison of UQ methods (QR, NGBoost, GPR, Deep Ensembles, MC Dropout, BNN-VI, BNN-MCMC) and reports standard metrics plus coverage and interval width. The results are consistent across datasets, and the authors are transparent about the limitations of BNN computational cost.

But there are load-bearing problems. First, Eq. (10) is written backwards. The second term ReLU(y*) penalizes positive predictions, so it pushes predictions negative, not non-negative. The third term ReLU(10,000,000 - y*) penalizes predictions below the upper bound, pushing them above it. If the code actually uses this loss, the boundary constraints are doing the opposite of what's claimed. If the code uses a corrected loss, then the paper misstates it. Either way, the physics-informed loss as described cannot be what improves results.

Second, Table 8 is identical to Table 4. Table 8 is supposed to be the torsion dataset PIML results, but the numbers match the uniaxial dataset. That's not a rounding coincidence; every metric matches. So either the tables are swapped or the experiment wasn't run properly.

Third, several NN rows have MAE > RMSE (e.g., Table 1 NN: RMSE 0.6443, MAE 2.0463). That's impossible for the same set of errors. It indicates the reported metrics don't come from the same predictions, or the errors were computed incorrectly.

Fourth, and most concerning, the Basquin feature is fit to the target data. The paper doesn't say whether the Basquin constants are fit on the training fold only or on the full dataset. Since the target is the experimentally recorded fatigue life, fitting on all data before the CV split means test-fold targets contribute to the features for that fold. That's target leakage, and it could explain the consistent 10–20% improvements across all models, including GPR and NGBoost which don't use the physics-informed loss. The stress-test note is right: the central claim is not established by the experiments as described.

The idea is still worth pursuing. A corrected version with a properly stated loss, a leakage-safe feature fitting procedure (fit Basquin constants inside each fold on training data only, and state it), and repaired tables could be a solid contribution. But as it stands, the paper doesn't support its headline claim.

My recommendation: if this lands on your desk, don't desk-reject it outright — send it to peer review with a clear request to verify the loss, the leakage protocol, and the tables. But warn the reviewers that the current version has serious inconsistencies, and the authors need to fix them before the paper can be accepted.

Referee Report

4 major / 4 minor

Summary. The paper proposes a physics-informed machine learning (PIML) framework for fatigue-life prediction with uncertainty quantification. The framework augments experimental fatigue datasets with a Basquin-model estimate of fatigue life as an additional input feature, and adds two boundary penalty terms to the loss function of neural-network models (non-negativity and an upper bound of 10,000,000 cycles). Experiments on one titanium-alloy dataset and three carbon-steel datasets (uniaxial, bending, torsion) compare seven regression models with and without the PIML additions, reporting point-estimate metrics (R2, PCC, RMSE, MAE) and uncertainty metrics (coverage, interval width, composite). The central claim is that the PIML additions improve both point estimates and uncertainty intervals across all four datasets.

Significance. If the reported improvements were valid, the approach would be a useful contribution to data-driven fatigue-life prediction with calibrated uncertainty, especially in small-data materials settings. The paper is strong in breadth: it compares several uncertainty-aware models (QR, NGBoost, GPR, Deep Ensemble, MC Dropout, BNN-VI, BNN-MCMC) on four datasets and makes code available. However, the significance is severely compromised because the core experimental evidence is undermined by an incorrect loss-term derivation, an undisclosed and potentially leaky feature-fitting protocol, a duplicated table, and internally inconsistent metric values. The current manuscript does not establish the claimed superiority of the PIML framework.

major comments (4)

[§3.3.2, Eq. (10)] The physics-informed loss terms enforce the opposite of the stated constraints. ReLU(y*) is zero for negative y* and penalizes positive predictions, thus driving predicted fatigue life toward zero or negative values rather than enforcing non-negativity; the correct term for a non-negativity constraint is ReLU(-y*). Similarly, ReLU(10,000,000 - y*) penalizes predictions below 10,000,000 and pushes predictions above the bound, whereas the intended upper-bound constraint requires ReLU(y* - 10,000,000). Additionally, the text states that fatigue-life values are log-transformed before training, so the bound of 10,000,000 cycles should also be transformed to the logarithmic scale if the loss is computed on log-transformed outputs. As written, Eq. (10) does not implement the claimed boundary constraints, and all NN-based PIML results that rely on this loss term are questionable.
[§3.3.1, §4.2] The procedure for fitting the Basquin constants c and m in Eq. (2) is not specified. The Basquin model is used to estimate fatigue life from experimental fatigue tests, which makes the resulting physics-informed feature a fitted function of the target variable. The paper does not state whether the constants are fitted inside each cross-validation fold, on the training folds only, or on the full dataset before the 5-fold split. If the fit uses the full dataset, the test-fold target values indirectly contribute to the training features, which is target leakage and could by itself explain the consistent 10-20% improvements across all models, including GPR and NGBoost that do not use the physics-informed loss. This must be clarified and the experiments must be rerun with fold-disciplined fitting before the central claim can be assessed.
[Tables 4 and 8] Table 8, labeled as the PIML results for the torsion fatigue test dataset of Carbon steel alloys, is numerically identical to Table 4, which reports the PIML results for the uniaxial fatigue test dataset. Every row and entry matches, including the standard deviations. This means the torsion-dataset PIML results are not actually presented, and the claim that the PIML approach improves results on all four datasets is not supported by the tables as published. This duplication must be corrected and the torsion experiments fully reported.
[Tables 1-8, NN rows] The reported mean absolute error (MAE) for the standard NN model is systematically larger than its root-mean-squared error (RMSE), e.g., Table 1 NN: MAE 2.0463 vs RMSE 0.6443; Table 3 NN: MAE 0.5999 vs RMSE 0.3982; Table 5 NN: MAE 0.5831 vs RMSE 0.3589. Since RMSE >= MAE for any fixed set of errors, these entries cannot be correct on the same scale. The NN metrics are therefore internally inconsistent, and the reported performance of the NN baseline (and its PIML variant) is not credible. These values must be recomputed and corrected, or the errors explained.

minor comments (4)

[§4.1] The definition of the composite metric is ambiguous in the text: '0.75 ∙ coverage + 0.25 mean interval width⁄' should be written as 0.75 * coverage + 0.25 / mean_interval_width, with the division made explicit.
[Eqs. (1) and (2)] The material constants in the S-N relation are called A and B in Eq. (1), while the Basquin relation in Eq. (2) uses c and m; the relationship between these parameters is not stated, which may confuse readers.
[Table 1, BNN-VI row] The composite metric for BNN-VI is reported as 0.7773±0.30; the large standard deviation relative to the mean is atypical for a bounded metric and may indicate an error in the reported aggregate.
[Throughout] There are several typographical issues, including 'Deep Ensemple' for 'Deep Ensemble' in the Table 1 discussion, and 'Mechanichs' for 'Mechanics' in reference titles. These should be corrected.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed PIML gain rests on a Basquin-estimated fatigue-life input feature that is itself fitted to the target fatigue-life data, with no fold-aware fitting protocol disclosed.

fitted input called prediction [Section 3.3.1 (Physics-informed Feature Engineering), using Eq. (2); target definition in Section 3.4; CV protocol in Section 4.2]
"Specifically, we employed the Basquin model from equation (2) to estimate the fatigue life of an alloy for a given stress level based on experimental data from fatigue tests. We used the estimated values for fatigue life as additional input features for training the ML models."

Eq. (2) is σ = c N_f^m; estimating N_f from it requires fitting c and m to experimental (σ, N_f) pairs and inverting to N_f = (σ/c)^(1/m). The target is exactly the measured fatigue life (Sec. 3.4), so the added feature is a fitted function of the target. The paper never states whether the Basquin fit is done inside each CV fold, on training folds, or on the full dataset, and reports no constants. If fit on all data before the split, each test sample's own fatigue life enters its input feature, so the consistent 10–20% PIML gains (Tables 2, 4, 6, and 8) can be explained by leakage. Even with training-fold-only fitting, the feature is target-derived rather than an independent physics constraint, so the central improvement claim is partially circular by construction.

full rationale

The only load-bearing circularity is in the physics-informed feature engineering. The Basquin feature is presented as physics knowledge but is computed by fitting the Basquin relation to the fatigue-life data that also defines the target, and no fold-safe fitting protocol or fitted constants are reported. This makes the headline PIML accuracy and uncertainty improvements statistically forced if the fit uses the full dataset. The physics-informed loss in Eq. (10) is not circular: it is a stated boundary regularizer applied only to NN models. The self-citation [18] (prior UQ work by overlapping authors) is used as background support and is not load-bearing for the PIML claim. Because the central claim reduces, as described, to feeding a target-derived fitted quantity into the regressors, the paper receives a partial circularity score; however, the underlying ML comparison framework and uncertainty metrics are independently defined, so the score is not at the top of the scale.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The core dependency is the fitted Basquin feature: the paper pulls the Basquin constants from the same target data and does not disclose the fitting protocol. The boundary loss adds two weights that are never reported. No new physical entities are introduced.

free parameters (2)

Basquin constants c and m = not reported
Used in Eq. (2) to compute the physics-informed fatigue-life feature in Section 3.3.1; the paper says they are estimated from experimental data but gives no values or fitting procedure.
Loss weights lambda1 and lambda2 = not reported
Weight the boundary terms in Eq. (10); no values or tuning procedure are given in Section 3.3.2.

assumptions (4)

domain assumption Basquin relation sigma = c * N_f^m (Eq. 2) adequately estimates fatigue life for the studied alloys at the stress levels present in the datasets.
Invoked in Section 3.3.1 to generate the added input feature; if Basquin is a poor fit, the feature injects systematic error.
domain assumption The Basquin constants can be estimated from the same experimental fatigue data used for training and testing without leaking target information.
Implied in Section 3.3.1; no cross-validation-aware fitting is described. If the constants are fit on the full dataset, test-fold target information enters the features.
domain assumption Fatigue life is bounded between 0 and 10,000,000 cycles, with 10,000,000 treated as the endurance-limit boundary.
Used to justify the penalty terms in Eq. (10), Section 3.3.2. The 10M threshold is a conventional choice, not a measured value for each material.
domain assumption Standard supervised-learning assumptions: the collected fatigue tests are independent and identically distributed samples representative of each alloy class.
Required for the 5-fold CV metrics in Section 4.2 to be meaningful; fatigue datasets are often censored or clustered by specimen batch, which is not addressed.

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Cite this review

Pith. "Pith review of Predictive Modeling and Uncertainty Quantification of Fatigue Life in Metal Alloys using Machine Learning." pith.science (2026). https://pith.science/paper/A3PBN62P

@misc{pith2026250115057,
  author       = {Pith},
  title        = {Pith review of: Predictive Modeling and Uncertainty Quantification of Fatigue Life in Metal Alloys using Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A3PBN62P}},
  note         = {Machine review of arXiv:2501.15057}
}

read the original abstract

Recent advancements in machine learning-based methods have demonstrated great potential for improved property prediction in material science. However, reliable estimation of the confidence intervals for the predicted values remains a challenge, due to the inherent complexities in material modeling. This study introduces a novel approach for uncertainty quantification in fatigue life prediction of metal materials based on integrating knowledge from physics-based fatigue life models and machine learning models. The proposed approach employs physics-based input features estimated using the Basquin fatigue model to augment the experimentally collected data of fatigue life. Furthermore, a physics-informed loss function that enforces boundary constraints for the estimated fatigue life of considered materials is introduced for the neural network models. Experimental validation on datasets comprising collected data from fatigue life tests for Titanium alloys and Carbon steel alloys demonstrates the effectiveness of the proposed approach. The synergy between physics-based models and data-driven models enhances the consistency in predicted values and improves uncertainty interval estimates.

Figures

Figures reproduced from arXiv: 2501.15057 by the authors.

**Figure 1.** High-level diagram of the proposed approach. The input parameters from a fatigue test dataset are augmented with physics-informed features and fed to a PIML model for predicting the fatigue life of a material. The data pre-processing steps include normalizing the input features to a range between 0 and 1, and applying logarithm transformation to the fatigue life values. To evaluate the models 5-fold crossvalidation… view at source ↗

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