Load-dependent Hardness Prediction for Materials using Machine Learning

Harikrishna Sahu; Madhubanti Mukherjee; Rampi Ramprasad

arxiv: 2604.20636 · v1 · submitted 2026-04-22 · ❄️ cond-mat.mtrl-sci · physics.app-ph

Load-dependent Hardness Prediction for Materials using Machine Learning

Madhubanti Mukherjee , Rampi Ramprasad , Harikrishna Sahu This is my paper

Pith reviewed 2026-05-09 23:58 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.app-ph

keywords machine learninghardness predictionVickers hardnessload dependenceexperimental datamaterials scienceDFT calculations

0 comments

The pith

Machine learning model using only experimental data and load value predicts hardness more accurately than ones mixing in computed moduli.

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

The paper builds machine learning models to predict Vickers hardness from a large set of experimental measurements that change with the force applied during testing. It shows that a model trained exclusively on those experimental results, treating the indentation load as an explicit input along with material composition and structure details, beats models that also draw on density functional theory calculations of elastic properties. This indicates that the load dependence of hardness cannot be fully replaced by bulk and shear moduli values. The result matters because superhard materials for wear applications require predictions that match real test conditions rather than idealized computations. The work underscores the need for experimental data that records measurement specifics.

Core claim

A single-task ML model trained solely on experimental Vickers hardness data, with explicit inclusion of indentation load along with compositional, electronic, and structural descriptors, outperforms multi-task models that combine experimental and DFT-computed data, showing that load is essential and sufficient for accurate hardness prediction beyond what bulk and shear moduli alone provide.

What carries the argument

Single-task machine learning regression model that treats indentation load as an input feature together with material descriptors.

If this is right

Reliable hardness models must treat the measurement load as a necessary variable rather than an optional detail.
Elastic moduli from computations cannot fully substitute for load-dependent experimental data in hardness prediction.
Hybrid models that add computed properties to experimental training do not improve accuracy when load is already included.
Screening workflows for hard materials should prioritize datasets that document indentation conditions.

Where Pith is reading between the lines

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

Hardness databases would become more useful if every entry recorded the exact load used in the test.
The same logic of explicit condition inclusion could improve predictions for other mechanical properties that vary with rate or load.
Models trained this way might be extended to forecast hardness at user-specified loads for particular engineering applications.

Load-bearing premise

The collected experimental hardness measurements form a representative dataset without systematic biases from test conditions or sample variations.

What would settle it

Applying both the single-task experimental model and a moduli-only model to a fresh collection of materials tested at multiple loads and finding that the experimental model no longer outperforms would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.20636 by Harikrishna Sahu, Madhubanti Mukherjee, Rampi Ramprasad.

**Figure 1.** Figure 1: FIG. 1. Hardness data showing (a) the distribution of experimental hardness values with respect to applied load and (b) the correlation between [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Performance of the single-task GPR model. (a) Learning [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

Superhard materials are critical for wear-resistant and high-stress applications. Conventional approaches correlating hardness with elastic moduli derived from DFT calculations enable rapid screening but overlook the strong load dependence of hardness. In this work, machine learning (ML) models were developed using a large, curated dataset of load-dependent experimental Vickers hardness (Hv) measurements. Moderate correlation was observed between experimental and DFT-based Hv values, whereas a single-task ML model trained solely on experimental data outperformed multi-task models that combined experimental and computed data. The superior performance of the single-task model highlights that explicit inclusion of indentation load, along with compositional, electronic, and structural descriptors, is essential and sufficient for accurate hardness prediction, beyond what can be achieved using DFT-accessible bulk and shear moduli alone (or in tandem with experimental data). These results emphasize the importance of high-quality experimental data and explicit inclusion of measurement conditions, particularly load, in the development of reliable hardness prediction models.

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.

Desk Editor's Note private letter to a colleague

Single-task ML on experimental load-dependent hardness data beats multi-task models with DFT, but the outperformance may trace to data handling rather than load itself.

read the letter

The main thing to know is that the authors built a dataset of experimental Vickers hardness values that include indentation load, then showed a single-task ML model trained only on that data outperforming multi-task versions that also incorporate DFT-computed bulk and shear moduli. They report moderate correlation between experimental and DFT hardness and conclude that explicit load plus compositional descriptors is enough, while DFT moduli add little once you have the experimental points with conditions attached. That comparison is the concrete new piece here. They do a solid job pulling together load-dependent experimental measurements, which most hardness ML papers skip, and they make a direct case that measurement conditions matter more than theory-derived elastic properties for practical screening. The emphasis on high-quality experimental ground truth over pure computation is fair and useful for the field. The soft spots sit in the details that are missing from the abstract and not fully resolved by the stress-test note. Dataset size, exact feature sets, how missing loads were treated in the multi-task case, loss weighting, and any ablation on the load feature itself are not reported, so it is hard to rule out that the single-task win comes from simpler training dynamics or domain shift rather than load being intrinsically essential. Moderate correlation alone does not strongly support the claim that load is sufficient beyond moduli. No error bars or significance tests are mentioned, which keeps the superiority claim provisional. This paper is for materials informatics groups working on mechanical properties who already have experimental data pipelines and want to see how load dependence changes the modeling picture. A reader focused on practical screening tools will get some value from the dataset curation and the exp-vs-DFT comparison, even if the methods need expansion. It is coherent enough on its own terms to deserve peer review, mainly to check the training setup and add the missing controls. I would send it to referees with a request for those specifics rather than desk-reject.

Referee Report

3 major / 2 minor

Summary. The paper develops machine learning models for predicting load-dependent Vickers hardness (Hv) of materials. It uses a curated dataset of experimental Hv measurements and reports moderate correlation with DFT-derived values. A single-task ML model trained solely on experimental data (with compositional, electronic, structural descriptors, and explicit indentation load) outperforms multi-task models that incorporate both experimental and computed (DFT) data. The authors conclude that explicit load inclusion is essential and sufficient for accurate predictions, beyond what bulk/shear moduli from DFT can achieve.

Significance. If the reported outperformance holds under rigorous validation, the work would usefully highlight limitations of purely DFT-based hardness screening and the value of incorporating measurement conditions like load. This could inform better ML practices for load-sensitive properties. However, the absence of key quantitative details (dataset size, architectures, validation protocols, and statistical tests) in the current manuscript limits assessment of whether the central claim is robust or an artifact of data partitioning and task setup.

major comments (3)

[Abstract] Abstract and Results: The claim that the single-task model 'outperformed' multi-task models is presented without any quantitative metrics (e.g., R², MAE, or RMSE values for each model, or statistical significance tests). This makes it impossible to evaluate the magnitude of the improvement or rule out that the gap arises from differences in hyperparameter tuning, loss weighting in multi-task training, or how the experimental subset is handled versus the DFT subset.
[Methods] Methods/Results: No details are provided on dataset size, composition (number of unique materials, load values per material), train/test split strategy, cross-validation procedure, or feature ablation studies isolating the contribution of the explicit load descriptor. Without these, it cannot be confirmed that the single-task superiority is specifically due to load inclusion rather than data-handling differences or domain shift between experimental and computed targets.
[Results] Results: The moderate experimental-DFT correlation is noted, but no comparison is shown between the single-task model and a baseline using only DFT moduli (without load) on the same experimental test set. This ablation is load-bearing for the claim that 'explicit inclusion of indentation load ... is essential and sufficient ... beyond what can be achieved using DFT-accessible bulk and shear moduli alone'.

minor comments (2)

[Methods] Clarify the exact definition and units of all descriptors (compositional, electronic, structural) and how indentation load is encoded as a feature (continuous value, binned, etc.).
[Results] Add a table summarizing model performance metrics across single-task, multi-task, and any DFT-only baselines for direct comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us identify areas where the manuscript can be strengthened. We agree that quantitative performance metrics, full dataset and methodological details, and an explicit DFT-moduli baseline comparison are necessary to robustly support our claims. We have revised the manuscript to incorporate all requested information and ablations. Below we respond point by point.

read point-by-point responses

Referee: [Abstract] Abstract and Results: The claim that the single-task model 'outperformed' multi-task models is presented without any quantitative metrics (e.g., R², MAE, or RMSE values for each model, or statistical significance tests). This makes it impossible to evaluate the magnitude of the improvement or rule out that the gap arises from differences in hyperparameter tuning, loss weighting in multi-task training, or how the experimental subset is handled versus the DFT subset.

Authors: We thank the referee for this observation. The revised manuscript now includes a new table in the Results section that reports R², MAE, and RMSE values for the single-task model, the multi-task models, and all relevant variants. We have also added the results of statistical significance tests (paired t-tests on the test-set predictions) to quantify the improvement. The Methods section has been expanded to describe the hyperparameter search procedure and the multi-task loss-weighting scheme, confirming that the performance gap is not an artifact of these choices. revision: yes
Referee: [Methods] Methods/Results: No details are provided on dataset size, composition (number of unique materials, load values per material), train/test split strategy, cross-validation procedure, or feature ablation studies isolating the contribution of the explicit load descriptor. Without these, it cannot be confirmed that the single-task superiority is specifically due to load inclusion rather than data-handling differences or domain shift between experimental and computed targets.

Authors: We agree that these details are essential for reproducibility and for isolating the role of the load descriptor. The revised Methods section now reports the total number of experimental data points, the number of unique materials, the distribution of indentation loads, the train/test partitioning strategy, and the cross-validation protocol. We have also added feature-ablation results that quantify the performance drop when the explicit load feature is removed while keeping all other descriptors fixed. These additions demonstrate that the single-task advantage is attributable to load inclusion rather than differences in data handling or domain shift. revision: yes
Referee: [Results] Results: The moderate experimental-DFT correlation is noted, but no comparison is shown between the single-task model and a baseline using only DFT moduli (without load) on the same experimental test set. This ablation is load-bearing for the claim that 'explicit inclusion of indentation load ... is essential and sufficient ... beyond what can be achieved using DFT-accessible bulk and shear moduli alone'.

Authors: We acknowledge that a direct comparison to a DFT-moduli-only baseline evaluated on the experimental test set is required to substantiate the claim. The revised Results section now includes this ablation: a model trained exclusively on DFT-derived bulk and shear moduli (no load information) and tested on the held-out experimental data. The load-inclusive single-task model outperforms this baseline, confirming that explicit incorporation of indentation load supplies predictive information beyond what is available from DFT moduli alone. A new figure illustrates the comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity in ML hardness prediction claims

full rationale

The paper reports empirical ML model results: single-task models trained only on experimental load-dependent Vickers hardness data outperform multi-task models that incorporate DFT-computed values. This outperformance is presented as an observed performance metric on held-out experimental data, not as a mathematical derivation. No equations are shown that equate predictions to inputs by construction, no parameters are fitted on a subset and then renamed as predictions, and no self-citations are invoked as load-bearing uniqueness theorems or ansatzes. The chain is self-contained because model training and evaluation follow standard supervised learning practices with experimental measurements as ground truth, independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the quality and representativeness of the experimental dataset, the sufficiency of chosen descriptors (compositional, electronic, structural, and load), and the assumption that ML performance differences reflect true predictive power rather than artifacts of training.

axioms (1)

domain assumption The curated experimental Vickers hardness dataset accurately represents material behavior under varying loads without significant measurement biases.
The paper treats this dataset as the basis for training and evaluation of all models.

pith-pipeline@v0.9.0 · 5465 in / 1175 out tokens · 29923 ms · 2026-05-09T23:58:19.200387+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Recent Advances in Superhard Materials , author=. Annu. Rev. Mater. Res , volume=

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Computational Discovery of Hard and Superhard Materials , author=. J. Appl. Phys. , volume=. 2019 , publisher=

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Computational materials science , volume=

Correlation between hardness and elastic moduli of the covalent crystals , author=. Computational materials science , volume=. 2011 , publisher=

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Hardness of covalent and ionic crystals: first-principle calculations , author=. Phys. Rev. Lett. , volume=

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Hardness of Covalent Crystals , author =. Phys. Rev. Lett. , volume =. 2003 , doi =

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Journal of Solid State Chemistry , volume=

Hard and superhard materials: a computational perspective , author=. Journal of Solid State Chemistry , volume=. 2019 , publisher=

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npj Computational Materials , volume=

Machine learning and evolutionary prediction of superhard BCN compounds , author=. npj Computational Materials , volume=. 2021 , publisher=

work page 2021

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Scientific reports , volume=

A statistical learning framework for materials science: application to elastic moduli of k-nary inorganic polycrystalline compounds , author=. Scientific reports , volume=. 2016 , publisher=

work page 2016

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Nature communications , volume=

Universal fragment descriptors for predicting properties of inorganic crystals , author=. Nature communications , volume=. 2017 , publisher=

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Journal of the American Chemical Society , volume=

Machine learning directed search for ultraincompressible, superhard materials , author=. Journal of the American Chemical Society , volume=. 2018 , publisher=

work page 2018

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npj Computational Materials , volume=

Predicting superhard materials via a machine learning informed evolutionary structure search , author=. npj Computational Materials , volume=. 2019 , publisher=

work page 2019

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Materials Letters , volume=

Machine learning guided discovery of super-hard high entropy ceramics , author=. Materials Letters , volume=. 2022 , publisher=

work page 2022

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Annals of Science , volume=

Hooke's Law and the Concept of the Elastic Limit , author=. Annals of Science , volume=. 1956 , publisher=

work page 1956

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Philosophical Magazine , volume=

On the relationships between hardness and the elastic and plastic properties of isotropic power-law hardening materials , author=. Philosophical Magazine , volume=. 2014 , publisher=

work page 2014

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Strength of Materials , volume=

Determination of Material Hardness Characteristics at the Elastic Limit by Instrumented Indentation , author=. Strength of Materials , volume=. 2025 , publisher=

work page 2025

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2011 , doi =

Modeling hardness of polycrystalline materials and bulk metallic glasses , journal =. 2011 , doi =

work page 2011

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2011 , doi =

Correlation between hardness and elastic moduli of the covalent crystals , journal =. 2011 , doi =

work page 2011

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1998 , author =

Computational Alchemy: The Search for New Superhard Materials , journal =. 1998 , author =

work page 1998

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and Gutekunst, Will R

Mukherjee, Madhubanti and Sahu, Harikrishna and Losego, Mark D. and Gutekunst, Will R. and Ramprasad, Rampi , title =. ACS Appl. Mater. Interfaces , volume =. 2024 , doi =

work page 2024

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and Day, Blake and Brgoch, Jakoah , title =

Zhang, Ziyan and Mansouri Tehrani, Aria and Oliynyk, Anton O. and Day, Blake and Brgoch, Jakoah , title =. Adv. Mater. , volume =. doi:https://doi.org/10.1002/adma.202005112 , year =

work page doi:10.1002/adma.202005112

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and Brgoch, Jakoah , title =

Hickey, Jacob C. and Brgoch, Jakoah , title =. Chem. Mater. , volume =. 2022 , doi =

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, title =

Jain, Anubhav and Ong, Shyue Ping and Hautier, Geoffroy and Chen, Wei and Richards, William Davidson and Dacek, Stephen and Cholia, Shreyas and Gunter, Dan and Skinner, David and Ceder, Gerbrand and Persson, Kristin A. , title =. APL Materials , volume =. 2013 , doi =

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work page 2015