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Materials-Discovery Workflows Guided by Symbolic Regression: Identifying Acid-Stable Oxides for Electrocatalysis

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

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Pith's one-line read An active-learning workflow using ensembles of SISSO symbolic-regression models finds 12 acid-stable oxides out of 1470 candidates in only 30 hybrid-DFT evaluations, versus 2 for random selection.

desk verdict A genuinely useful SISSO-based active-learning workflow with a real screening application, but the headline efficiency claim rests on a threshold mismatch that needs fixing. read the letter →

arxiv 2412.05947 v1 pith:TY7WHNPX submitted 2024-12-08 cond-mat.mtrl-sci cs.AI

Akhil S. Nair , Lucas Foppa , Matthias Scheffler This is my paper · ORCID
classification cond-mat.mtrl-scics.AI
keywords activelearningsymbolicregressionSISSOacid-stableoxideselectrocatalysisPourbaixdecompositionenergyuncertaintyquantificationmaterialsdiscovery
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

This paper argues that active-learning materials discovery does not need to know the few governing parameters in advance, because symbolic regression can find them on the fly. The authors build ensembles of SISSO models, adding Monte-Carlo dropout of input features to ordinary bagging, and show that this reduces prediction error and tames the overconfidence of single SISSO or bagged models. Using the ensemble's probability of feasibility as the acquisition function, the workflow finds 12 acid-stable oxides for electrocatalytic water splitting out of 1470 candidates in just 30 high-cost hybrid-DFT evaluations, whereas random selection finds 2. If the approach holds, high-quality but expensive calculations can be spent where they matter, and the resulting descriptor maps give a physical explanation for the search.

What carries the argument

The load-bearing mechanism is an ensemble construction the paper calls bagging with Monte-Carlo dropout of primary features: each ensemble member is a SISSO model trained on a bootstrap sample of the data with a random 20% subset of the 14 primary features retained. Averaging these models gives the mean prediction $\Delta G_{\text{pbx,ESISSO}}$, and their spread gives the uncertainty estimate $\sigma_{\text{ESISSO}}$ used in the acquisition function $\text{POF} = F\!\left(\frac{\tau - \Delta G_{\text{pbx,ESISSO}}}{\sigma_{\text{ESISSO}}}\right)$, the Gaussian cumulative probability that a candidate is acid-stable. Together they turn SISSO into a closed-loop discovery engine that selects one oxide per iteration for hybrid-DFT evaluation and retrains.

What would settle it

A decisive test would be to rerun the same 30-iteration campaign with an acquisition function that only exploits the ensemble mean, ignoring the uncertainty term; if the mean-only strategy also finds 12 acid-stable oxides, then the calibrated uncertainty estimates are not essential to the claimed gain.

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Extended reading notes

Core claim

The central claim is that bagging with Monte-Carlo dropout of primary features creates SISSO ensembles whose mean prediction and standard deviation are good enough to steer active learning. On a held-out test set the ensemble reaches a mean absolute error of 0.26 eV/atom, down from 0.34 for a single SISSO model and 0.29 for plain bagging, and its miscalibration score drops from 2.76 for bagging to 1.76, still overconfident but markedly improved. The acquisition function, probability of feasibility, is the Gaussian cumulative probability that a candidate's Pourbaix decomposition free energy lies below the stability threshold. In 30 iterations this strategy identifies 12 acid-stable oxides, several of which are missed by PBE-level screening, and the SISSO descriptor map shows the selected materials clustering in the low-energy region.

Load-bearing premise

The ranking of candidates by probability of feasibility assumes that each prediction error is Gaussian with standard deviation equal to the ensemble's spread, and that this spread reliably orders which candidates the model knows least about; the paper's own calibration analysis shows the ensemble is still overconfident, so the search's efficiency rests on this unproven ranking property.

Editorial extensions

If this is right

  • If correct, hybrid-DFT screening of oxide stability becomes practical for thousands of materials instead of dozens.
  • The same workflow can be applied to other properties whose governing parameters are unknown, as long as primary features and an ensemble of symbolic-regression models can be defined.
  • The descriptor maps produced by SISSO give interpretable axes for the materials space, showing which chemical motifs, such as Mo, Ta, and W, favor acid stability.
  • Because five of the twelve discovered oxides are misclassified by PBE, the result implies that cheaper exchange-correlation functionals can mislead high-throughput screens and that ensemble uncertainty flags where higher-level calculations are needed.
  • The workflow reduces the number of expensive calculations by roughly a factor of six relative to random selection in this pool, a direct measure of its efficiency.

Reading between the lines

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

  • I infer that Monte-Carlo feature dropout works because it prevents the ensemble from over-agreeing on a single descriptor set; a similar effect could be obtained by sampling over the SISSO hyperparameters q and D, which the paper does not test.
  • I infer that the POF criterion will be most valuable in early iterations when the surrogate is least certain; after the descriptor space is well mapped, pure exploitation may become nearly as efficient, an effect visible in the shrinking error bars in the paper's Figure 2.
  • I infer that the 12 oxides, which are rich in Mo, Ta, and W, suggest a design rule worth testing in the laboratory: acid stability at pH 0 and 1.23 V correlates with these oxophilic early-transition-metal frameworks, and the descriptor map could be used to propose new ternary compositions.
  • I infer that the overconfidence remaining, with a mean z of 1.76, means the workflow's hit rate could be improved further by replacing the Gaussian cumulative-distribution assumption in POF with a heavier-tailed or nonparametric calibration, a modification the paper leaves open.
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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 develops an active-learning (AL) workflow based on ensembles of SISSO symbolic-regression models. To quantify prediction uncertainty, the authors compare three ensemble strategies (bagging, model-complexity bagging, and bagging with Monte-Carlo dropout of primary features) and find that the latter yields the lowest MAE and the most balanced miscalibration scores on held-out data. They then use this ensemble in an AL campaign with a probability-of-feasibility (POF) acquisition function to discover acid-stable oxides from a pool of 1470 candidates, using DFT-HSE06 as the ground-truth evaluator. In 30 AL iterations, they report identifying 12 acid-stable oxides, whereas random selection finds only 2. They also provide SISSO-derived descriptor maps of oxide stability and compare PBE and HSE06 predictions for the discovered materials.

Significance. If the findings are robust, this is a valuable demonstration of symbolic regression in closed-loop materials discovery, going beyond interpolation-based AL by offering interpretable descriptors and uncertainty estimates. The ensemble comparison is carefully done with 30 independent trials, and the public availability of code, data, and a tutorial strengthens reproducibility. The practical result—discovering acid-stable oxides that are missed by GGA-based screening—is of interest to the electrocatalysis community. However, the central efficiency claim rests on several choices that are not fully justified, notably the threshold τ used in the acquisition function and the lack of repeated AL campaigns; these need to be addressed to establish the quantitative advantage claimed.

major comments (3)
  1. [Results (POF definition)] Equation (1) defines the acquisition probability with τ = 0.00, but the paper labels an oxide as acid-stable only if ΔG_pbx^OER ≤ 0.1 eV/atom (stated in the paragraph after Eq. (1) and in Figure 2). Since σ_ESISSO varies across candidates, this threshold mismatch does not merely shift all POF values by a constant; it changes the ranking of candidates and therefore the set of materials acquired in each AL iteration. The reported efficiency gain (12 acid-stable oxides found by POF vs. 2 by random selection in 30 iterations) is thus contingent on an arbitrary threshold choice. The authors should either justify τ = 0, rerun the AL campaign with τ = 0.1, or at least show that the conclusions are insensitive to τ over a plausible range.
  2. [Results (AL campaigns)] The central claim that POF-guided AL identifies 12 acid-stable oxides in 30 iterations is based on a single AL campaign. The ensemble comparison in Figure 1 is repeated over 30 independent trials, but the AL results in Figure 2 are not; given the stochasticity of bootstrap sampling and Monte-Carlo feature dropout, a different random seed could yield a different trajectory and a different number of discoveries. The authors should report the distribution of outcomes over multiple campaigns (e.g., 5–10 seeds) or at least provide a sensitivity analysis with respect to the initial training set and the random seed.
  3. [Results (uncertainty calibration)] The ensemble method used for acquisition has a mean miscalibration score z = 1.76, indicating overconfident uncertainty estimates. Equation (1) treats σ_ESISSO as a Gaussian standard deviation, but the calibration analysis shows that POF values are not true probabilities. The paper acknowledges this limitation but does not test whether the AL efficiency actually derives from the uncertainty term. A useful control would be an acquisition strategy based only on the ensemble mean prediction (e.g., selecting the lowest predicted ΔG_pbx^OER) to separate the exploitation and exploration contributions; if that baseline performs comparably to POF, the uncertainty-driven exploration claim would need to be moderated.
minor comments (4)
  1. [Figure 2] The filled and open square markers are difficult to distinguish in grayscale; consider using different shapes or colors with accessible palettes.
  2. [Results (POF definition)] The acronym 'POF' is defined as 'probability of feasibility', but the term 'feasibility' is not standard for a stability threshold; consider 'probability of stability' or define the target event more explicitly.
  3. [Methods (SISSO)] In the operator set in Eq. (2), the division φ1/φ2 is included but there is no statement about how zero denominators are handled; please clarify this implementation detail.
  4. [Results (initial dataset)] The composition and selection criteria of the initial 250-oxide training set are deferred to the Supplementary Material; a brief statement in the main text would help the reader judge potential bias in the training distribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SISSO-guided discovery claim is validated by independent DFT-HSE06 calculations, not by the fitted surrogate.

full rationale

The paper's derivation chain is self-contained against an external benchmark. SISSO is a surrogate fitted to DFT-HSE06 values of ΔG_pbx, and ensembles provide mean predictions and uncertainties used by the POF acquisition strategy. Crucially, the acid-stable materials reported are not taken from SISSO predictions; they are confirmed by direct DFT-HSE06 Pourbaix analysis, as stated: 'the selected oxides along with computed acid stability are added to the dataset' and Figure 2 compares SISSO predictions against DFT-HSE06 values. The central claim of 12 acid-stable oxides in 30 iterations is therefore an empirical outcome of the closed-loop workflow, not a consequence of fitting or self-citation. The SISSO-related citations are to published methods and code (e.g., refs. [13], [15], [39]) that are externally validated and not target-specific; they do not smuggle in the result. No equation in the paper reduces to its own inputs by construction. The noted mismatch between the POF threshold τ=0.00 and the acid-stability threshold ΔG≤0.1 eV/atom is a potential acquisition-objective alignment concern, but it is not circularity, because the final discovery count is determined by DFT calculations, not by the POF score. Overall, the paper's logic does not exhibit any of the seven circularity patterns, so the score is 0.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claim depends on hand-chosen hyperparameters (q, D, k, dropout), on the assumption that HSE06 Pourbaix free energy is the right definition of acid stability, on the unstated Gaussian error model in POF, and on the representativeness of the initial training and candidate data. The paper does not introduce new physical entities.

free parameters (6)
  • SISSO rung q = 2
    Maximum operator depth; chosen via nested cross-validation (Supplementary Section IV). This controls the complexity of generated analytical expressions.
  • SISSO descriptor dimension D = 2
    Number of descriptor components; selected by nested cross-validation.
  • Ensemble size k = 10
    Number of bootstrap models; chosen based on convergence of prediction errors and uncertainty (Figure S11).
  • Feature dropout fraction = 0.2 (20% retained)
    Fraction of primary features used in each ensemble member; set to 20% without a reported systematic optimization.
  • POF threshold tau = 0.00 eV/atom
    Threshold in Eq. (1) for acid-stable in acquisition; differs from the 0.1 eV/atom labeling threshold, a mismatch not discussed in the text.
  • Acid-stability labeling threshold = 0.1 eV/atom
    Used to label an oxide as acid-stable; justified by self-passivation, but the POF target uses 0.0 eV/atom, so the acquisition is not aligned with the final count.
assumptions (4)
  • domain assumption DFT-HSE06 Pourbaix decomposition free energy is an accurate proxy for acid stability under OER conditions (pH=0, 1.23 V).
    The entire discovery target is this computed quantity; no experimental validation is provided. The paper motivates HSE06 over GGA in Supplementary I-III but does not benchmark against measurements.
  • domain assumption The candidate space of 1470 oxides and the initial 250 training oxides are representative enough for the AL comparison.
    The composition and selection of the candidate space is deferred to Supplementary Section V; if the space is biased toward unstable materials or excludes relevant stable phases, the efficiency comparison holds only for this space.
  • ad hoc to paper Prediction errors of the SISSO ensemble are Gaussian with mean and variance from the ensemble, as assumed in Eq. (1).
    The POF formula uses the standard normal CDF without testing normality; the authors' own z-scores show miscalibration (z>1), so this is a working assumption rather than an established fact.
  • domain assumption SISSO's selected descriptors generalize across the materials space (extrapolation via physical parameters).
    The workflow relies on the SISSO model trained on initial data to rank candidate materials; the paper argues symbolic forms generalize better than interpolation, but this is not proven and some discovered materials were initially mispredicted stable/unstable.

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Cite this review

Pith. "Pith review of Materials-Discovery Workflows Guided by Symbolic Regression: Identifying Acid-Stable Oxides for Electrocatalysis." pith.science (2026). https://pith.science/paper/TY7WHNPX

@misc{pith2026241205947,
  author       = {Pith},
  title        = {Pith review of: Materials-Discovery Workflows Guided by Symbolic Regression: Identifying Acid-Stable Oxides for Electrocatalysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TY7WHNPX}},
  note         = {Machine review of arXiv:2412.05947}
}
read the original abstract

The efficiency of active learning (AL) approaches to identify materials with desired properties relies on the knowledge of a few parameters describing the property. However, these parameters are unknown if the property is governed by a high intricacy of many atomistic processes. Here, we develop an AL workflow based on the sure-independence screening and sparsifying operator (SISSO) symbolic-regression approach. SISSO identifies the few, key parameters correlated with a given materials property via analytical expressions, out of many offered primary features. Crucially, we train ensembles of SISSO models in order to quantify mean predictions and their uncertainty, enabling the use of SISSO in AL. By combining bootstrap sampling to obtain training datasets with Monte-Carlo feature dropout, the high prediction errors observed by a single SISSO model are improved. Besides, the feature dropout procedure alleviates the overconfidence issues observed in the widely used bagging approach. We demonstrate the SISSO-guided AL workflow by identifying acid-stable oxides for water splitting using high-quality DFT-HSE06 calculations. From a pool of 1470 materials, 12 acid-stable materials are identified in only 30 AL iterations. The materials property maps provided by SISSO along with the uncertainty estimates reduce the risk of missing promising portions of the materials space that were overlooked in the initial, possibly biased dataset.

Figures

Figures reproduced from arXiv: 2412.05947 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Schematic representation of different [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: SISSO-descriptor-based material maps of oxide [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Materials Database from All-electron Hybrid Functional DFT Calculations

    cond-mat.mtrl-sci 2025-04 accept novelty 6.0 of 10

    An open database of 7,024 inorganic materials computed with all-electron HSE06 hybrid DFT, including stability metrics and a SISSO model for HSE06 band gaps.

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