-->PONE-D-26-17436-->-->Physics-Informed Gaussian Process Regression for Reproducible and Uncertainty-Aware CO2 Injectivity Prediction-->-->PLOS One

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Reviewer #1: Partly

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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-->5. Review Comments to the Author

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Reviewer #1: The manuscript presents a timely and methodologically rigorous investigation into physics-informed Gaussian process regression for predicting CO2 injectivity decline, addressing critical gaps in uncertainty quantification, reproducible evaluation, and physical constraint enforcement within geological carbon storage applications. The integration of DLVO colloidal monotonicity and the Civan–Kozeny–Carman permeability impairment model into a Gaussian process framework represents a thoughtful advancement over purely data-driven approaches, and the implementation of a three-tier validation protocol combining leave-one-out cross-validation, repeated k-fold cross-validation, and non-parametric bootstrap confidence intervals establishes a commendable standard for statistical robustness in small-sample geoscience studies. The authors’ emphasis on uncertainty-aware modeling, particularly through the combination of GP posterior calibration using Expected Calibration Error and split-conformal prediction intervals, is highly valuable for operational decision-making in carbon capture and storage, where risk quantification is as crucial as point prediction accuracy. The operational risk atlas translating predictive uncertainty into actionable injection scheduling guidance effectively bridges the gap between laboratory-scale modeling and field deployment, and the transparent acknowledgment of the accuracy–uncertainty trade-off inherent in physics-informed modeling at limited data densities demonstrates scholarly maturity. However, despite these substantial strengths, the manuscript requires major revision to address several methodological, contextual, and evaluative shortcomings before it can meet the publication standards of PLOS ONE. The most pressing concern revolves around the extremely limited sample size of n=44 laboratory measurements, which inherently restricts the statistical power and generalizability of all reported models. While the authors appropriately employ rigorous resampling techniques to mitigate overfitting, the fundamental constraint of learning complex nonlinear interactions across four primary input variables and eight engineered features from fewer than fifty observations warrants more cautious interpretation of the reported R2 values and confidence intervals. The manuscript would benefit significantly from a more thorough discussion of the bias-variance trade-off under such severe data scarcity, particularly regarding how the Matérn-5/2 kernel hyperparameter optimization might be influenced by sparse regions in the high-salinity, high-jamming-ratio regime. Furthermore, the reliance on a single curated laboratory dataset from prior literature introduces potential batch effects and experimental design limitations that are not fully accounted for in the uncertainty propagation framework; a sensitivity analysis demonstrating how model performance degrades under systematic perturbations or subsampling strategies would strengthen the claim of robustness. Regarding the physics-informed components, the observation that the unconstrained GP-Base model achieves superior leave-one-out accuracy and calibration compared to the constrained variants is an important empirical finding, yet the underlying mechanism driving this result requires deeper theoretical exposition. The authors attribute the slight performance degradation in PC-GPR-M and PC-GPR-MC to constraint tension and reduced posterior flexibility, but a more rigorous analysis of how virtual derivative observations and prior mean functions interact with the GP marginal likelihood optimization process would be highly instructive. Specifically, the manuscript should clarify whether the 1.5% DLVO violation rate in PC-GPR-M stems from insufficient inducing point density, suboptimal virtual observation strength, or inherent kernel incompatibility, and provide ablation studies isolating the contribution of each constraint parameter. Additionally, the claim that domain-guided feature engineering implicitly encodes DLVO monotonicity, while empirically supported by the zero violation rate in GP-Base, should be contextualized within the broader literature on implicit versus explicit physics integration, particularly regarding how engineered interaction terms like Bidx and S·Q approximate underlying pore-scale electrostatic and hydrodynamic mechanisms without guaranteeing thermodynamic consistency. Enhance connections to recent literature that demonstrates the great potential and use of several machine learning models, such as the neural network (doi: 10.1142/S2737599425500367), Gaussian process regression (doi: 10.1080/17509653.2025.2453902; 10.1007/s00521-024-10726-w), and ensemble/composite method (doi: 10.1002/ajae.12041), for modelling complicated (nonlinear) patterns across a broad variety of study subjects in order to further motivate the exploration of machine learning models in your present work. Building upon this contextual expansion, the manuscript’s comparative analysis should explicitly address why Gaussian process regression was selected over other modern probabilistic or hybrid architectures that have demonstrated comparable or superior uncertainty quantification capabilities in similar geoscientific domains. While the O(n3) computational cost is manageable for n=44, the authors should discuss scalability considerations and potential alternatives such as sparse variational Gaussian processes or deep kernel learning, which could better accommodate future experimental expansions. The baseline model selection, comprising Linear Regression, Bayesian Ridge, grid-search SVR, and a stacking ensemble, is adequate but somewhat dated; incorporating more contemporary benchmarks such as gradient-boosted decision trees with calibrated outputs or neural network ensembles would provide a more comprehensive performance landscape. Moreover, the treatment of the published GA-SVR benchmark as a single-split upper-tail outcome is statistically sound, yet the manuscript would be strengthened by a formal statistical test comparing the distribution of cross-validated R2 scores across all models rather than relying solely on visual inspection of confidence interval overlap. Use or at least discuss other performance measures, such as the relative root mean square error, that facilitate performance comparisons across models or targets (see doi: 10.1007/s43674-024-00075-5). In conjunction with expanding the evaluation metrics, the authors should provide a more nuanced interpretation of the Average Absolute Percentage Error values, which appear disproportionately high relative to the reported R2 scores, suggesting potential scale-dependency issues or outlier sensitivity that warrants explicit diagnostic testing. The conformal prediction framework is correctly implemented and provides valuable finite-sample coverage guarantees, but the manuscript lacks a discussion on conditional versus marginal coverage, which is particularly relevant given the heterogeneous uncertainty structure across the salinity-jamming ratio space. Operators deploying this model in field conditions would benefit from understanding whether the 97.73% nominal coverage holds uniformly across different operational regimes or is concentrated in low-salinity, low-jamming regions where training data are denser. Furthermore, the operational risk atlas, while conceptually excellent, should be accompanied by quantitative decision thresholds or cost-benefit analyses that translate prediction interval widths into concrete injection rate adjustments or economic risk metrics, thereby enhancing its practical utility for reservoir engineers. The technical presentation of the methodology is generally clear, but several aspects of the implementation require greater transparency to ensure full reproducibility. The manuscript should specify the exact random seeds, optimization algorithms, and convergence criteria used for hyperparameter tuning, particularly regarding the eight random restarts for log-marginal likelihood maximization and the multi-start L-BFGS-B optimization for the Civan prior parameters. The feature engineering process, while physically motivated, introduces derived variables that may induce multicollinearity, and the authors should report variance inflation factors or correlation matrices to demonstrate that robust scaling adequately mitigates numerical instability in the GP kernel computations. Additionally, the code availability statement references an MT-AAE repository, which appears unrelated to the Gaussian process framework described in the manuscript; this discrepancy must be corrected to point to the actual CO2 injectivity prediction repository, and all preprocessing pipelines, constraint implementations, and conformal calibration routines should be fully documented. The limitations section appropriately acknowledges the sparse experimental grid, the modest standalone explanatory power of the Civan prior, and the absence of field-scale validation, but these constraints should be more rigorously integrated into the main discussion rather than relegated to a brief subsection. The manuscript would significantly benefit from a prospective experimental design recommendation specifying how additional laboratory measurements should be strategically allocated across the input space to maximize information gain, particularly in the high-uncertainty region identified at salinity > 30,000 ppm and jamming ratio > 0.04. Furthermore, the coupling of rate-dependent deposition hysteresis and wettability alteration into the mechanistic prior is mentioned as future work, but the current formulation’s inability to capture these dynamics should be explicitly quantified through residual analysis, demonstrating how much unexplained variance remains attributable to unmodeled physical processes versus stochastic experimental noise. In summary, while the manuscript delivers a novel, uncertainty-aware, and physically constrained modeling framework that advances the state of the art in CO2 injectivity prediction, it requires substantial revisions to strengthen its theoretical grounding, expand its comparative and evaluative scope, clarify its implementation details, and enhance its practical relevance for subsurface engineering applications. Addressing these concerns through methodological refinements, broader literature contextualization, expanded performance diagnostics, and corrected reproducibility artifacts will significantly elevate the manuscript’s scientific rigor and operational impact. I recommend major revision to allow the authors the opportunity to comprehensively address these points, after which the work will be well-positioned for publication as a robust and reproducible contribution to physics-informed machine learning in geological carbon storage.

Reviewer #2: The manuscript addresses an important practical problem of uncertainty‑aware prediction of CO2 injectivity and makes a valuable contribution by introducing conformal prediction and uncertainty quantification to this domain. The three‑tier validation is a step forward from single‑split reporting. However, the paper overstates the benefits of physics constraints given the small dataset and lack of extrapolation validation, and the comparison with the GA‑SVR benchmark is not methodologically sound. The authors should justify or make less strong claims about “guarantees” and “operational risk maps”, clarify the limitations of the small dataset, and add extrapolation experiments to support the claimed advantage of constrained models. With these revisions, the work would be a valuable resource for the CCS and geoscience machine learning communities.

1) The dataset size (n=44) is extremely small for generalising the conclusions. The finding that GP‑Base naturally respects DLVO monotonicity (0% violations) without explicit constraints is interesting, but with only 44 points, the posterior is heavily influenced by the kernel prior and the feature engineering. The authors should provide a sensitivity analysis to show whether the same conclusions hold when n is varied, or at least temper the claims about “physical interpretability of the engineered feature set” as tentative.

2) The Civan prior mean function has very low standalone predictive power (R2=0.299). While a low‑accuracy prior can still be useful, the manuscript does not quantify how much the GP posterior improves upon this prior in the constrained variants. The paper should include a dedicated extrapolation test to demonstrate the claimed advantage of the physics‑constrained models.

3) The benchmark comparison with GA‑SVR is methodologically flawed. The authors compare their GP models (trained and evaluated on the same 44‑point dataset) with the GA‑SVR results reported by Mardhatillah et al. (2022). However, the GA‑SVR was evaluated on a single 80/20 split, while the GP models are evaluated under cross‑validation. The statement that the GA‑SVR value (0.9923) falls within the upper tail of the GP‑Base repeated k‑fold CI [0.643,0.989] is not a valid criticism because the CI is computed on the GP’s cross‑validated R2, not on a single‑split test R2. Cross‑validated R2 is generally lower than a favourable single‑split test R2 even for the same model. To make a fair comparison, the authors should either: (i) train the GA‑SVR model themselves on the same data and evaluate it under the same three‑tier protocol, or (ii) compute a single‑split test R2 for GP‑Base on the exact same 80/20 split used by Mardhatillah et al. and then compare.

4) The “physics‑informed” claim is overstated for PC‑GPR‑M and PC‑GPR‑MC. The DLVO monotonicity constraint is enforced through virtual derivative observations, which is a soft constraint (1.5% violations remain). The authors call this a “DLVO physical compliance guarantee”, but a guarantee would require hard constraints (e.g., via constrained GP formulations or transformation of the input space). The manuscript should clarify that the constraints are softand that the violation rate is reported to characterise the degree of compliance, not a strict guarantee.

5) Feature engineering (12 features from 4 raw inputs) risks overfitting. With n=44 and d=12, the ratio of samples to features is only about 3.7:1. While the robust IQR scaling and GP kernel hyperparameter optimisation are sensible, the authors do not report any feature selection or dimensionality reduction. The high LOO R2 of linear regression (0.965) suggests that the feature set is nearly collinear or that the problem is almost linear in the transformed space.

6) The operational risk atlas is based on extrapolation into sparsely sampled regions. The high‑uncertainty region (S > 30,000 ppm, J > 0.04) contains only four observations. The authors present this as an actionable risk map for injection scheduling. However, without any field validation or independent test points in that region, the map is essentially a visualisation of the GP posterior variance, not a validated risk assessment. The manuscript should clearly state that the risk atlas is illustrative and should not be used for operational decisions without further experimental validation.

7) The three‑tier validation protocol is not applied consistently to the baselines. The baselines (LR, BR, SVR‑GS, Stack) are evaluated under LOO and repeated k‑fold, but they do not have GP‑specific metrics like ECE or conformal coverage. This is fine. However, the authors claim that the protocol “sets a minimum reporting standard for future ML studies”. To be a standard, the protocol should be agnostic to model class; the authors should demonstrate that it can be applied to any regression model, including the baselines, without modification. The current implementation does not achieve this. The authors should either extend conformal prediction to all models or rename the protocol to “GP‑specific validation”.

Minor issues:

8) The literature review is currently limited. I suggest extending it by referring to more recent publications in physics‑informed machine learning and geoscience applications, such as https://www.nature.com/articles/s41598-024-65954-w and https://www.nature.com/articles/s41598-026-50929-w

9) The “1.3 Contributions” section begins directly with a list of what has been done. I suggest starting with a brief description of the goals and the specific problem solved by the authors, and only then presenting the list of main contributions as outcomes of the conducted research.

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Reviewer #1: No

Reviewer #2: No

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Physics-Informed Gaussian Process Regression for Reproducible and Uncertainty-Aware CO2 Injectivity Prediction

PONE-D-26-17436R1

Dear Dr. Adamu,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Reviewers' comments:

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Reviewer #1: (No Response)

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Reviewer #1: (No Response)

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Reviewer #1: (No Response)

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Reviewer #1: (No Response)

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Reviewer #1: (No Response)

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-->6. Review Comments to the Author

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Reviewer #1: The authors have adequately addressed all of my previous comments. I recommend that the manuscript be accepted for publication in its current form.

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Reviewer #1: No

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