pith. sign in

arxiv: 2606.21948 · v1 · pith:JOEPTM4Tnew · submitted 2026-06-20 · 🧮 math.OC

Active learning-based Bayesian optimization in the realm of copper slag-blended cement systems

Debadri Som , Rayna Maheshwari , Mathijs Schuurmans , Panagiotis Patrinos , Yiannis Pontikes This is my paper

Pith reviewed 2026-06-26 11:51 UTC · model grok-4.3

classification 🧮 math.OC
keywords Bayesian optimizationcement mix designcopper slaglow-carbon cementGaussian processactive learningmulti-objective optimization
0
0 comments X

The pith

Bayesian optimization locates low-emission copper slag cement blends meeting strength targets after 2-6 additional tests beyond an initial 10.

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

The paper demonstrates that Bayesian optimization, initialized with only 10 experiments at a 2:1 data-to-variable ratio, can sequentially select new mix designs for copper slag, limestone, and Portland cement systems. It targets 32.5N strength class while balancing 2-day compressive strength, cost, and CO2 emissions using Gaussian process models updated with conservative uncertainty bounds for 28-day strength. A sympathetic reader would care because conventional mix design campaigns demand far more laboratory trials, and this method promises faster identification of formulations that stay below 500 kg CO2 per ton while satisfying industry strength minima.

Core claim

Starting from 10 initial experiments, Gaussian process surrogate models with linear and RBF kernels, driven by Expected Improvement and Upper Confidence Bound acquisition functions, identify Pareto non-dominated solutions within 2-6 iterations that deliver greater than 10 MPa at 2 days, greater than 32.5 MPa at 28 days, and under 500 kg CO2/ton; linear kernels show higher predictive accuracy while RBF kernels provide better uncertainty estimates for active learning.

What carries the argument

Bayesian optimization loop that fits Gaussian process surrogates to compressive strength, cost, and emissions data and uses acquisition functions to choose the next mix proportion to test under laboratory data constraints.

If this is right

  • Optimal blends for the target strength class can be reached with far fewer total experiments than traditional grid or factorial designs.
  • A conservative update rule that incorporates uncertainty bounds allows iteration without completing every 28-day test before selecting the next point.
  • Linear kernels outperform RBF kernels on point prediction accuracy in this material system, yet RBF kernels are better suited for guiding the search via uncertainty.
  • The resulting formulations align with current industry targets for cement decarbonization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sequential selection strategy could shorten development cycles for other supplementary cementitious materials whose response surfaces are comparably smooth.
  • Embedding the acquisition-function step directly into automated dosing equipment would further compress the time between suggestion and measurement.
  • Extending the objective set to include additional durability metrics would test whether the same data-efficiency advantage persists.

Load-bearing premise

Gaussian process models with the chosen kernels can accurately represent the true response surfaces for strength and emissions in the copper slag-limestone-cement system even when trained on only 10 points.

What would settle it

Prepare and test the BO-recommended blends at full scale and measure whether their 28-day compressive strength exceeds 32.5 MPa and their CO2 emissions stay below 500 kg per ton.

read the original abstract

Accelerated mix design optimization is critical for deploying low-carbon supplementary cementitious materials (SCMs) because traditional experimental approaches require extensive testing campaigns. The authors demonstrate that Bayesian optimization (BO) can identify near-optimal blended cement formulations using an AI-driven approach with minimal experimental data. Starting with only 10 initial experiments and a 2:1 data-to-variable ratio reflecting realistic laboratory constraints, the authors optimized copper slag, limestone, Portland cement systems for 32.5N strength class with respect to 2-day compressive strength, cost, and CO2 emissions. Gaussian process surrogate models guided sequential experimentation through Expected Improvement and Upper Confidence Bound acquisition functions. Within 2-6 iterations, BO identified Pareto non-dominated solutions meeting strength requirements (greater than 10 MPa at 2 days, greater than 32.5 MPa at 28 days) while achieving CO2 emissions below 500 kg CO2/ton; consistent with industry decarbonization targets. A conservative update strategy incorporating uncertainty bounds for 28-day strength enabled rapid iteration without waiting 28 days per cycle. Comparative analysis revealed that linear kernels outperformed nonlinear alternatives in predictive accuracy, though radial basis function kernels were preferred for active learning due to superior uncertainty quantification. This work demonstrates BO as a practical decision-support tool for cement research under severe data constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies Bayesian optimization with Gaussian process surrogates to optimize copper slag-limestone-Portland cement blends for 32.5N strength class. Starting from 10 initial experiments (2:1 data-to-variable ratio), it uses Expected Improvement and Upper Confidence Bound acquisition functions to identify Pareto-optimal formulations meeting >10 MPa at 2 days and >32.5 MPa at 28 days while keeping CO2 emissions below 500 kg/ton, all within 2-6 iterations. A conservative update rule handles 28-day strength feedback; linear kernels are reported to outperform others in accuracy while RBF kernels are preferred for uncertainty quantification.

Significance. If the experimental validation holds, the work demonstrates a practical, data-efficient decision-support tool for low-carbon cement mix design under realistic lab constraints (limited initial data, delayed 28-day curing). Strengths include grounding in actual experimental measurements rather than simulation, explicit handling of the 28-day feedback delay via conservative bounds, and a direct kernel comparison that informs surrogate choice for active learning in this domain. These elements address a genuine industrial need for accelerated SCM deployment.

major comments (2)
  1. [Results section (kernel comparison and iteration outcomes)] The central claim that BO identified near-optimal solutions meeting all targets rests on the surrogate models accurately capturing the response surfaces of strength, cost, and emissions. However, the abstract supplies no cross-validation scores, hold-out prediction errors, or uncertainty calibration metrics for the GP models trained on the initial 10 points; without these in the results or methods, it is impossible to confirm that the 2-6 iteration convergence is not an artifact of model misspecification.
  2. [Methods (acquisition functions and update rule)] The conservative 28-day update strategy is load-bearing for the rapid iteration claim, yet no quantitative description is given of how uncertainty bounds were propagated into the acquisition function or how many 28-day measurements were ultimately performed to close the loop. This detail is required to evaluate whether the reported Pareto solutions are experimentally confirmed or only predicted.
minor comments (2)
  1. [Abstract and Results] The abstract states linear kernels 'outperformed nonlinear alternatives in predictive accuracy' without naming the metric (RMSE, MAE, or log-likelihood) or reporting the numerical values; add a table or sentence with these numbers.
  2. [Methods] Notation for the acquisition functions (EI and UCB) and the precise definition of the multi-objective Pareto front should be introduced consistently in the methods before being used in the results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the practical relevance of our work and for the constructive major comments. We address each point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Results section (kernel comparison and iteration outcomes)] The central claim that BO identified near-optimal solutions meeting all targets rests on the surrogate models accurately capturing the response surfaces of strength, cost, and emissions. However, the abstract supplies no cross-validation scores, hold-out prediction errors, or uncertainty calibration metrics for the GP models trained on the initial 10 points; without these in the results or methods, it is impossible to confirm that the 2-6 iteration convergence is not an artifact of model misspecification.

    Authors: We agree that explicit quantitative metrics for the initial GP models are necessary to substantiate the surrogate quality and the reported convergence. While the manuscript already presents a kernel comparison based on predictive accuracy, we will add a dedicated subsection in Results with leave-one-out cross-validation scores, hold-out prediction errors (MAE/RMSE), and uncertainty calibration plots (e.g., coverage of 95% intervals) for the models trained on the 10 initial points. These additions will directly address the concern and allow readers to evaluate whether the 2-6 iteration outcomes are supported by model fidelity. revision: yes

  2. Referee: [Methods (acquisition functions and update rule)] The conservative 28-day update strategy is load-bearing for the rapid iteration claim, yet no quantitative description is given of how uncertainty bounds were propagated into the acquisition function or how many 28-day measurements were ultimately performed to close the loop. This detail is required to evaluate whether the reported Pareto solutions are experimentally confirmed or only predicted.

    Authors: We acknowledge that the current Methods description of the conservative update rule is qualitative. We will expand this section to provide the precise mathematical formulation for propagating the GP uncertainty bounds (upper/lower confidence intervals) into the acquisition functions, including the specific conservative threshold applied. We will also report the exact number of 28-day compressive strength measurements performed across the campaign and clarify which of the final Pareto solutions were experimentally validated versus those relying on the conservative prediction. This revision will make the experimental closure of the loop fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies standard Bayesian optimization with Gaussian processes and acquisition functions (EI, UCB) to a cement formulation problem. It starts from 10 initial experiments and updates sequentially based on new lab data for 2-day strength, with a conservative rule for 28-day estimates. No derivation reduces a claimed prediction to a fitted parameter by construction, no self-citation chain justifies a uniqueness result, and no ansatz is smuggled in. The linear vs. RBF kernel comparison and Pareto front identification rest on external experimental outcomes rather than internal redefinition. This is a conventional engineering application of BO under data constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides insufficient detail to identify any free parameters, axioms, or invented entities used in the work.

pith-pipeline@v0.9.1-grok · 5782 in / 1232 out tokens · 39147 ms · 2026-06-26T11:51:40.072869+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 4 canonical work pages

  1. [1]

    Initialization of data set: Set 𝑡 = 0, 𝒟𝑡 = 𝒟0

  2. [2]

    GP models: Define GPs for 2-day and 28-day strength with 𝜇 as the mean and 𝑘 as the kernel function (section 2.3.1): 𝑓2(𝑥) ∼ 𝒢𝒫(𝜇2, 𝑘2) 𝑓28(𝑥) ∼ 𝒢𝒫(𝜇28, 𝑘28)

  3. [3]

    3.2 Normalization: 𝜇̃2(𝑥) = 𝜇2(𝑥)−𝜇2,min 𝜇2,max−𝜇2,min , 𝑐̃(𝑥) = 𝑐̂(𝑥)−𝑐min 𝑐max−𝑐min , 𝑒̃(𝑥) = 𝑒̂(𝑥)−𝑒min 𝑒max−𝑒min

    Bayesian Optimization (BO) loop: while stopping criterion not met (defined in section 2.2.3) 3.1 Training: Fit GPs on 𝒟𝑡 to obtain (𝜇2, 𝜎2), (𝜇28, 𝜎28). 3.2 Normalization: 𝜇̃2(𝑥) = 𝜇2(𝑥)−𝜇2,min 𝜇2,max−𝜇2,min , 𝑐̃(𝑥) = 𝑐̂(𝑥)−𝑐min 𝑐max−𝑐min , 𝑒̃(𝑥) = 𝑒̂(𝑥)−𝑒min 𝑒max−𝑒min

  4. [4]

    Scalarization (section 2.2.2): 𝑠(𝑥) = 𝑤1𝜇̃2(𝑥) − 𝑤2𝑐̃(𝑥) − 𝑤3𝑒̃(𝑥) 𝜎̃𝑠(𝑥) = 𝑤1𝜎2(𝑥) 𝜇2,max−𝜇2,min 𝑠∗ = max 𝑥𝑖∈𝒟𝑡 𝑠(𝑥𝑖); where 𝑠∗ is best scalar 3.4 Sampling: Generate N candidates 𝑥𝑗 ∼ 𝒰(𝒳) satisfying mass balance. 3.5 Prediction: Evaluate 𝜇2(𝑥𝑗), 𝜎2(𝑥𝑗), 𝜇28(𝑥𝑗), 𝜎28(𝑥𝑗) 3.6 Acquisition (section 2.3.2): compute Expected Improvement (EI) or Upper Confiden...

  5. [5]

    Return: 𝒟𝑡 and ranked feasible designs 𝒟𝑟 2.3.1 Gaussian process modelling Referring to point 3 in the algorithm, GPs were used to train the models for 2 -day and 28-day compressive strength. Gaussian Processes (GPs) are non -parametric models widely used as surrogate models for expensive -to-evaluate functions, particularly in the context of Bayesian opt...

    2006

  6. [6]

    The length-scale combinations suggested by the grid search, along with the corresponding train and test 𝑅2values, are summarized in Table 4

    Results and Discussions 3.1 Kernel performance and selection Based on the grid search-based length-scale optimization of kernels , the radial basis function (RBF) kernel provided the best overall predictive performance among the kernels considered. The length-scale combinations suggested by the grid search, along with the corresponding train and test 𝑅2va...

    2023

  7. [7]

    The conclusions are structured to directly address the research question s posed in this work

    Conclusions This study investigated the applicability of BO as a data -efficient framework for multi - objective blended cement mix design under practical experimental and compositional constraints. The conclusions are structured to directly address the research question s posed in this work. • Effectiveness of BO under limited data BO successfully identi...

    2026

  8. [8]

    They have been mentioned in the points below which are classified broadly into four categories

    Recommended directions for future work There are several areas that needs further work in the context of applying BO for cement and concrete research. They have been mentioned in the points below which are classified broadly into four categories

  9. [9]

    However, the kernels used are extremely critical in ensuring a stable performing GP model useful as a surrogate in the BO loop

    GP models and kernels: GP models tend to work well as a surrogate model in BO and has been used extensively in BO literature. However, the kernels used are extremely critical in ensuring a stable performing GP model useful as a surrogate in the BO loop. In this context, categorical features should be considered differently as opposed to the current and pr...

    2024

  10. [10]

    This is expected to increase the chances of finding a global optimum in the BO loop

    Acquisition function optimization: In place of sampling points and selecting on the basis of maximized acquisition functions, the acquisition function can itself be maximized using a hybrid solver (Noack and Funke, 2017) which combines a local and a global solver which can efficiently negotiate multiple local minima without taking a long time to converge....

    2017

  11. [11]

    This will eventually help in benchmarking techniques and identifying best practices in the context of a powerful technique like BO for cement and concrete research

    Practical applications: The techniques (or similar techniques) discussed in this paper should be applied more in research in place of traditional techniques, wherever possible. This will eventually help in benchmarking techniques and identifying best practices in the context of a powerful technique like BO for cement and concrete research. There are autom...

    2025

  12. [12]

    However, important durability characteristics like freeze -thaw, acid attack and others should be included in the process of decision-making

    Objective functions: In this study, the focus was primarily on compressive strength, cost and CO 2 equivalent. However, important durability characteristics like freeze -thaw, acid attack and others should be included in the process of decision-making. However, in those cases, testing times are usually long (up to 12 months at times) which defeats the pur...

    2020

  13. [13]

    Optimization of zeolite replacement in concrete mix design based on mechanical, economic, and environmental criteria using machine learning and metaheuristic algorithms,

    References Afshan, A.G. and Ramezanianpour, A.M. (2026) “Optimization of zeolite replacement in concrete mix design based on mechanical, economic, and environmental criteria using machine learning and metaheuristic algorithms,” Engineering Applications of Artificial Intelligence , 172, p. 114343. Available at: https://doi.org/10.1016/j.engappai.2026.11434...

    work page doi:10.1016/j.engappai.2026.114343 2026

  14. [14]

    Hydration of blended cement with high volume iron -rich slag from non-ferrous metallurgy,

    Available at: https://doi.org/10.1007/s40831-025-01129-9. Hallet, V . et al. (2022) “Hydration of blended cement with high volume iron -rich slag from non-ferrous metallurgy,” Cement and Concrete Research , 151, p. 106624. Available at: https://doi.org/10.1016/j.cemconres.2021.106624. Hallet, V . et al. (2023) “The hydration of ternary blended cements wit...

    work page doi:10.1007/s40831-025-01129-9 2022

  15. [15]

    Future and emerging supplementary cementitious materials,

    Available at: https://doi.org/10.1016/j.mfglet.2024.09.157. Snellings, R., Suraneni, P. and Skibsted, J. (2023) “Future and emerging supplementary cementitious materials,” Cement and Concrete Research , 171, p. 107199. Available at: https://doi.org/10.1016/j.cemconres.2023.107199. Som et al. (2026) “Data for strength, cost and GWP of copper slag-blended c...

    work page doi:10.1016/j.mfglet.2024.09.157 2024

  16. [16]

    Sundar, X

    Available at: https://doi.org/10.1557/s43578-024-01489-0. Tam, V .W.Y . (2022) “A prediction model for compressive strength of CO2 concrete using regression analysis and artificial neural networks,” Construction and Building Materials [Preprint]. Available at: https://doi.org/https://doi.org/10.1016/j.conbuildmat.2022.126689. US Department of Energy (2023...

    work page doi:10.1557/s43578-024-01489-0 2022