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REVIEW 3 major objections 4 minor 27 references

Neural Network-Guided Symbolic Regression for Interpretable Descriptor Discovery in Perovskite Catalysts

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-phase neural-network-guided symbolic regression pipeline can discover accurate, interpretable descriptors for oxygen evolution reaction activity in oxide perovskites, finding that a formula using μ/t, μ/RA…

desk verdict A clearly described two-phase pipeline that honestly reproduces the known μ/t descriptor, but the LUMO claim rests on validation-set reuse and the reported improvement is noise. read the letter →

arxiv 2507.12404 v1 pith:FLCRA5DH submitted 2025-07-16 physics.data-an cond-mat.mtrl-scics.LGphysics.comp-ph

Yeming Xian , Xiaoming Wang , Yanfa Yan This is my paper · ORCID
classification physics.data-ancond-mat.mtrl-scics.LGphysics.comp-ph
keywords oxygenevolutionreactionoxideperovskitessymbolicregressionneuralnetworkfeatureimportanceLUMOenergydescriptordiscoverysmall-datamachinelearningoverpotentialprediction
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 claims that a two-phase machine-learning pipeline can find compact, interpretable formulas for oxygen evolution reaction (OER) activity in oxide perovskites even when only 23 samples are available. In the first phase, a neural network ranks a small set of structural descriptors and engineered composite features such as μ/t and μ/RA; symbolic regression then recovers and slightly improves the known μ/t descriptor. In the second phase, the feature space is expanded to 164 candidates, reduced to 46, and the neural network identifies LUMO energy as a new electronic descriptor. A symbolic formula combining μ/t, μ/RA, and LUMO energy predicts overpotential with training and validation mean absolute errors of 22.1 and 20.6 meV, respectively. If correct, this means interpretable descriptors can be extracted without sacrificing accuracy in data-scarce materials problems.

What carries the argument

The machinery is a two-phase pipeline. A regularized three-layer neural network is trained on the available samples, and permutation feature importance—computed by shuffling each feature and measuring the drop in validation performance—identifies which descriptors matter. Composite features are then engineered from the top structural descriptors, and in the expanded phase the descriptor space is filtered by variance and correlation thresholds before symbolic regression searches over candidate formulas. The central object that carries the final result is the three-descriptor formula $\frac{\mu}{t}+\frac{\mu}{R_A}+E_{LUMO}+(-E_{LUMO})^{0.5}+0.780$, interpreted through an optimal LUMO window that reflects intermediate binding strength on the catalyst surface.

What would settle it

Re-run the pipeline with the five validation samples completely excluded from feature ranking and formula selection, or with leave-one-out cross-validation across all 23 samples, and check whether a LUMO-containing formula is still chosen and whether its out-of-sample error stays near 20.6 meV. Independently, compare the composition-weighted atomic LUMO proxy against measured or density-functional-theory-calculated solid-state LUMO levels for the same perovskites; if the proxy does not track the true LUMO, the proposed electronic interpretation would need revision.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that OER overpotential for oxide perovskites can be described by an explicit formula, $\frac{\mu}{t}+\frac{\mu}{R_A}+E_{LUMO}+(-E_{LUMO})^{0.5}+0.780$, where $\mu$ is the octahedral factor, $t$ the tolerance factor, $R_A$ the A-site ionic radius, and $E_{LUMO}$ a composition-weighted estimate of the lowest unoccupied molecular orbital energy. The LUMO term is not treated as a mere fitting variable: measured overpotential is non-monotonic in LUMO energy, with the best samples falling in an optimal window around $-0.34$ to $-0.32$ eV, which the authors explain through the Sabatier principle as balanced adsorption and desorption of reaction intermediates. The same workflow reproduces and improves the earlier μ/t rule when restricted to seven structural features, and the expanded-feature phase shows that the LUMO term can be discovered from compositional information alone.

Load-bearing premise

The load-bearing assumption is statistical: the same five held-out samples are used to rank features, to select the best of 432 symbolic formulas, and then to report the final validation error, so the 20.6 meV figure assumes this procedure does not bias the estimate.

Editorial extensions

If this is right

  • If the central claim is correct, LUMO energy becomes a physically meaningful electronic descriptor for OER activity in oxide perovskites, not just a statistical correlate.
  • The formula gives a concrete screening rule: favor low μ/t, low μ/RA, and LUMO energies in roughly the -0.34 to -0.32 eV window to minimize overpotential.
  • The two-phase approach demonstrates a general recipe for small-data descriptor discovery: use a neural network to cut the feature space before symbolic regression, rather than running symbolic regression on all features.
  • Phase I independently confirms and slightly improves the previously reported μ/t descriptor, suggesting the pipeline can recover known physics before finding new descriptors.

Reading between the lines

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

  • The paper leaves untested whether the 20.6 meV validation error is optimistic, because the same five validation samples participate in both feature ranking and formula selection; an independent test set or nested validation would be needed to certify the number.
  • Because the LUMO feature is a composition-weighted average of atomic LUMO values rather than a computed solid-state level, the apparent electronic descriptor may inherit its predictive power from electronegativity and valence-orbital information; testing against explicit electronic-structure calculations would show whether the mechanism is genuinely electronic.
  • The framework is likely to transfer to other data-scarce structure-property problems, but its main advantage over direct symbolic regression should shrink as sample sizes grow and plain symbolic regression becomes more stable.
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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 two-phase machine-learning workflow for discovering interpretable symbolic formulas that predict oxygen evolution reaction (OER) activity (VRHE) of oxide perovskites from a small dataset (23 samples). Phase I uses seven structural features, adds engineered composites (μ/t, μ/RA, RA·t), and derives a formula (μ/RA)^0.5 + μ/t + 0.775 with reported training and validation MAEs of 22.8 and 20.8 meV. Phase II expands to 164 Matminer features, filters to 46, identifies LUMO energy as a top descriptor via permutation importance, and reports a final formula μ/t + μ/RA + E_LUMO + (−E_LUMO)^0.5 + 0.780 with training and validation MAEs of 22.1 and 20.6 meV, claiming improved accuracy and strong physical interpretability. The paper also benchmarks the neural network against Ridge, random forest, XGBoost, and PCA+XGBoost.

Significance. If the statistical foundation were sound, this paper would be a valuable demonstration of combining neural-network feature ranking with symbolic regression for data-scarce materials problems. The workflow is clearly described, the benchmark against traditional ML models is useful, and the idea of an electronic descriptor alongside structural descriptors is interesting. However, the core claim - that LUMO is a genuine descriptor and that the Phase II formula achieves improved validation accuracy - rests on a statistical protocol that reuses the same five-sample validation set for feature selection, formula selection, and final performance reporting. The reported validation MAEs are therefore better interpreted as selections, not predictions. The physical interpretation of LUMO is also not independently validated, as it uses an atomic composition-weighted proxy rather than a solid-state electronic level. With explicit acknowledgment of these limitations or additional validation, the work could be reframed as a promising hypothesis-generation pipeline, but the current claims are not supported by the evidence presented.

major comments (3)
  1. [Methods, 'Neural Network Training'] The permutation feature importance analysis used to select LUMO as a key descriptor was performed on the same five-sample validation set that later serves as the evidence for the final formula's accuracy (Methods: 'The analysis was performed on the validation set using R² as the scoring metric'). This creates a selection-dependence loop: the validation set is used to rank features, then to choose the symbolic-regression formula, and then to report the headline validation MAE of 20.6 meV. With n = 5, a 0.2 meV difference between Phase I and Phase II validation MAEs is far smaller than the sampling uncertainty of the mean absolute error, so the abstract's claim of 'improved accuracy' for the Phase II formula is not supported. The paper should either report an independent holdout (e.g., leave-one-out or nested cross-validation that includes the feature and formula selection steps) or clearly label the reported validation numbers as selection results rather than generalization estimates.
  2. [Results, 'Refined Formula and Performance'] The final formula was selected from 432 gplearn hyperparameter configurations by validation MAE, and among the two formulas that contained all three features, the one with lower validation MAE (0.0206 vs. 0.0283 eV, Table S3) was chosen. Because the same five validation points were used for this selection, the reported validation MAE is an optimistically biased estimate of the formula's performance on unseen materials. No correction for this selection bias is provided, and Table S3 does not report any confidence interval or per-point error distribution. The claim that the formula 'achieves the lowest validation error' is therefore not evidence that it generalizes; at minimum, the paper must quantify the bias (e.g., through a bootstrap over the selection procedure) or treat the formula as a hypothesis to be tested on new data.
  3. [Results, 'Physical Interpretability of LUMO'] The paper interprets the LUMO feature as a solid-state electronic energy level that controls binding of OER intermediates via the Sabatier principle (Figure 5b). However, as the text itself states, the feature is Matminer's AtomicOrbitals LUMO, a composition-weighted average of atomic LUMO levels rather than a solid-state eigenvalue. No electronic-structure calculation is performed to support the claim that this atomic proxy corresponds to the perovskite LUMO level. The non-monotonic relationship in Figure 5a is based on 23 points and an optimal window (-0.34 to -0.32 eV) that is inferred from the same data, so the mechanistic narrative is post hoc. The identification of LUMO as a 'key electronic descriptor' is thus not established; the authors should either validate the proxy against first-principles calculations or moderate the claim to say that the composition-weighted atomic LUMO is a promising empirical feature.
minor comments (4)
  1. [Supplementary Tables S1 and S2] Several entries in Tables S1 and S2 contain typographic artifacts such as repeated superscripts (e.g., '0.50.5') and missing parentheses (e.g., formula I), making the formulas hard to read; please regenerate the tables with a proper equation renderer.
  2. [Figure 5d] The caption refers to 'the single green point' without defining the color scheme for the points; please add a legend or explain the color mapping in the caption.
  3. [Discussion, 'Interpretation of Descriptors'] The sentence 'suggesting to increase t, decrease μ, and adjust components to acquire optimal LUMO energy' is incomplete as a design rule; clarify how the formula's coefficients imply these recommendations, given the non-monotonic LUMO term.
  4. [Methods, 'Symbolic Regression with gplearn'] The description of the hyperparameter grid states p_crossover ranged from 0.5 to 0.925 in increments of 0.025, but it does not specify how the remaining probability mass is split among the three mutation operators (subtree, hoist, point); please provide the exact grid, as the reproducibility of the 432-formula search depends on it.

Circularity Check

2 steps flagged · score 6.0 of 10

Validation-set reuse in feature and formula selection makes the reported validation MAE a selection statistic, so the LUMO descriptor claim is not independently established.

  1. fitted input called prediction [Methods, Neural Network Training; Results, Phase II – Expanded Feature Space (New Features and Importance)]
    "We used permutation feature importance from scikit-learn to identify the most predictive features. The analysis was performed on the validation set using R² as the scoring metric, with 30 random shuffles per feature to ensure stability. ... Permutation feature importance analysis (Figure 4c) revealed a new electronic descriptor—LUMO energy—as one of the most influential features in the model. ... Its prominence in the NN model suggests a strong link between electronic structure and catalytic performance, and supports its inclusion in the symbolic regression step to follow."

    The five-sample validation set is used to compute permutation importance, which is the step that identifies LUMO as a key descriptor. That same descriptor is then included in the final symbolic formula, and the formula is evaluated on the same validation set. The validation labels therefore influenced which feature was selected, so the subsequent validation performance is not an independent prediction; the reported 'discovery' of LUMO reduces to the fact that permuting LUMO on those five validation points hurt the validation R². The Methods explicitly state that the importance analysis was performed on the validation set, making the selection and the evaluation share the same data.

  2. fitted input called prediction [Results, Phase II – Refined Formula and Performance; Table S3]
    "Among the formulas generated across the symbolic regression hyperparameter space, only two candidates incorporated all three features. Their structures, lengths, and associated MAEs on training and validation sets are summarized in Table S3. The better-performing formula achieved a training MAE of 0.0221 eV and a validation MAE of 0.0206 eV, slightly improving upon the Phase I models while maintaining a compact symbolic form."

    The final formula is selected from the 432-candidate symbolic-regression search by its validation MAE: only two candidates contained all three features, and 'the better-performing formula' is chosen by comparing validation MAEs. Reporting that chosen formula's validation MAE as evidence of improved accuracy treats the selection objective as if it were an unbiased holdout estimate. With only five validation points, selecting the better of two formulas (themselves generated from a 432-formula search) makes the 20.6 meV figure a fitted quantity rather than an independent prediction, so the headline 'improved accuracy' claim is partially circular.

full rationale

The main derivation chain is not self-contained because the same five-sample validation set is used both to select features/formulas and to report the headline error. The Methods state that permutation importance—the basis for choosing LUMO as a key descriptor—was computed on the validation set, and the Results show that the final Phase II formula was selected as the better of two candidates by validation MAE. The reported 20.6 meV validation MAE is therefore a model-selection statistic, not an unbiased estimate of generalization; the 0.2 meV improvement over Phase I is within the noise of a five-point MAE. The training MAE (22.1 meV) is a genuine in-sample fit, and the Phase I baseline against Weng et al. is an external benchmark, so the paper is not wholly circular. But the central claim that LUMO is an important electronic descriptor is supported only by a validation-set-dependent importance ranking, and the optimal LUMO window is read from the same 23 samples, so the 'prediction' partially reduces to the selection loop.

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

The primary uncontrolled inputs are the fitted formula constants and the hand-set feature filtering thresholds; the most consequential assumption is that the Matminer atomic-orbital LUMO proxy stands in for the true solid-state LUMO in the physical narrative. The paper invents no new entities: LUMO comes from an existing featurizer, and the optimal LUMO window is a data-derived range, not a postulated object.

free parameters (5)
  • Constant term in Phase II formula = 0.780
    Appears in the final formula μ/t + μ/RA + E_LUMO + (−E_LUMO)^0.5 + 0.780; fitted by the symbolic regressor to minimize MAE on the 18-sample training set.
  • Constant term in Phase I formula = 0.775
    Appears in (μ/RA)^0.5 + μ/t + 0.775; fitted by gplearn on the 18-sample training set.
  • Feature filtering thresholds = variance ≥ 1e-4; Pearson correlation ≤ 0.95
    Hand-chosen thresholds reduce 164 Matminer features to 46; which features survive to the NN, including LUMO, depends on this choice.
  • Symbolic regression hyperparameter grid = crossover 0.5-0.925 step 0.025; parsimony 0.0005-0.0015; population 5000; 20 generations
    Grid search over 432 settings; the setting that produced the reported formulas is not disclosed, and selection among candidates used validation MAE.
  • Neural network architecture and training choices = 128-64-16 ReLU, L2 regularization, Adam, early stopping with warm-up
    Chosen without a reported tuning protocol; permutation importance inherits these choices and the model is trained on 23 samples.
assumptions (4)
  • domain assumption The 23 OER overpotential measurements from Weng et al. are accurate and representative.
    All training and validation numbers inherit the source dataset; the paper performs no new measurements or uncertainty propagation.
  • domain assumption Matminer's AtomicOrbitals LUMO, a composition-weighted average of atomic LUMO estimates, is a meaningful proxy for the perovskite's solid-state LUMO level.
    Invoked in 'Physical Interpretability of LUMO' to justify the descriptor's role in intermediate binding; no band-structure or DFT validation is given.
  • domain assumption The Sabatier principle explains the observed non-monotonic trend in VRHE versus LUMO energy.
    The optimal-binding story is presented as the mechanism behind the data-derived optimal LUMO window; it is an interpretation, not a tested prediction.
  • standard math gplearn, scikit-learn permutation importance, TensorFlow, and Matminer implementations behave as documented.
    The workflow relies on these libraries without modification or formal verification.

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Cite this review

Pith. "Pith review of Neural Network-Guided Symbolic Regression for Interpretable Descriptor Discovery in Perovskite Catalysts." pith.science (2026). https://pith.science/paper/FLCRA5DH

@misc{pith2026250712404,
  author       = {Pith},
  title        = {Pith review of: Neural Network-Guided Symbolic Regression for Interpretable Descriptor Discovery in Perovskite Catalysts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLCRA5DH}},
  note         = {Machine review of arXiv:2507.12404}
}
read the original abstract

Understanding and predicting the activity of oxide perovskite catalysts for the oxygen evolution reaction (OER) requires descriptors that are both accurate and physically interpretable. While symbolic regression (SR) offers a path to discover such formulas, its performance degrades with high-dimensional inputs and small datasets. We present a two-phase framework that combines neural networks (NN), feature importance analysis, and symbolic regression (SR) to discover interpretable descriptors for OER activity in oxide perovskites. In Phase I, using a small dataset and seven structural features, we reproduce and improve the known {\mu}/t descriptor by engineering composite features and applying symbolic regression, achieving training and validation MAEs of 22.8 and 20.8 meV, respectively. In Phase II, we expand to 164 features, reduce dimensionality, and identify LUMO energy as a key electronic descriptor. A final formula using {\mu}/t, {\mu}/RA, and LUMO energy achieves improved accuracy (training and validation MAEs of 22.1 and 20.6 meV) with strong physical interpretability. Our results demonstrate that NN-guided symbolic regression enables accurate, interpretable, and physically meaningful descriptor discovery in data-scarce regimes, indicating interpretability need not sacrifice accuracy for materials informatics.

Figures

Figures reproduced from arXiv: 2507.12404 by the authors.

Figure 2
Figure 2. (a) Neural network architecture used for Phase I/II modeling. The network consists of three hidden layers with 128, 64, and 16 neurons respectively (ReLU activation and L2 regularization), and a linear output layer predicting OER activity (VRHE). This architecture was used to identify important features and guide symbolic regression. (b) Neural network performance. Predicted versus measured VRHE values for the train… view at source ↗
Figure 4
Figure 4. (a) Feature expansion and reduction workflow. Matminer generated 164 numerical features, which were reduced to 46 using variance and Pearson correlation filters. Original and composite features were retained throughout. (b) Predicted versus measured VRHE using the neural network trained on the reduced 46-feature dataset. (c) Permutation feature importance for the Phase II neural network model. Among the newly added … view at source ↗
Figure 5
Figure 5. (a) Relationship between LUMO energy and measured VRHE for 23 oxide perovskite samples. (b) Schematic illustration of the variance of OER activity as a result of LUMO level fluctuation with respect to the Fermi energy. (c) Predicted versus measured VRHE using the refined symbolic regression formula involving μ/t, μ/RA, and LUMO energy. (d) 3D visualization of measured VRHE as a function of μ/t, μ/RA, and LUMO energy… view at source ↗

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Reviewed August 6, 2026 · model on record in the stance chip above.