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arxiv: 2506.07656 · v1 · submitted 2025-06-09 · 🧮 math.NA · cs.NA· math.OC

Data-Informed Mathematical Characterization of Absorption Properties in Artificial and Natural Porous Materials

Elishan C. Braun , Gabriella Bretti , Melania Di Fazio , Laura Medeghini , Mario Pezzella This is my paper

Pith reviewed 2026-05-19 10:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC
keywords porous materialswater imbibitionabsorption characterizationmonotonicity-preserving fittingpartial differential equationscultural heritage preservation
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The pith

A monotonicity-preserving fitting procedure paired with PDE calibration characterizes water absorption properties in porous materials such as marble and mortar.

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

The paper presents laboratory imbibition tests on mock-ups of marble, travertine, wackestone, and mortar to measure water absorption. A new fitting procedure is introduced that smooths the data while preserving its monotonic increase to reduce noise and errors. The imbibition is modeled using a partial differential equation whose parameters are calibrated using both the original measurements and the smoothed versions. Results indicate that this approach effectively distinguishes the absorption behaviors of the different materials tested.

Core claim

Experimental data from imbibition tests on marble, travertine, wackestone and mortar mock-ups are used to inform and validate the mathematical and simulation frameworks. A monotonicity-preserving fitting procedure is developed to preprocess the measurements, and the imbibition process is simulated through a partial differential equation model with parameters calibrated against rough and smoothed data. The procedure effectively characterizes absorption properties of different materials.

What carries the argument

The monotonicity-preserving fitting procedure that preprocesses noisy absorption measurements while keeping the increasing trend, used to calibrate a PDE model of the imbibition process.

If this is right

  • The method allows reliable comparison of absorption properties across natural and artificial porous materials.
  • It supports the development of simulation tools for predicting water movement in these materials.
  • It serves as a tool for the study and preservation of cultural heritage objects composed of such materials.

Where Pith is reading between the lines

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

  • This method may apply to characterizing absorption in other porous substances like concrete or biological tissues.
  • Further validation could involve applying the calibrated models to predict absorption under different environmental conditions not tested in the original experiments.
  • Combining this approach with material imaging could enable more detailed spatial analysis of porosity effects.

Load-bearing premise

Smoothing the data with the monotonicity-preserving procedure does not distort the actual physical absorption rates used in the PDE calibration.

What would settle it

New absorption experiments on the same material samples where the PDE predictions from parameters fitted to smoothed data deviate significantly from observations, unlike those from raw data.

Figures

Figures reproduced from arXiv: 2506.07656 by Elishan C. Braun, Gabriella Bretti, Laura Medeghini, Mario Pezzella, Melania Di Fazio.

Figure 1
Figure 1. Figure 1: Examples of natural stones and mortars in ancient buildings and archaeological sites. texture and exhibits color variations caused by mineral impurities. The travertine poros￾ity is extremely variable and dependent on the deposit it comes from [31, 47]. With the term wackestone is defined, according to the Dunham and Folk classification systems of limestones [1], a fine-grained mud-supported carbonate sedi… view at source ↗
Figure 2
Figure 2. Figure 2: Materials samples employed for the laboratory capillary ab￾sorption tests. Material Porosity range N Reference Carrara Marble 0.50% − 4.00% [38] Tivoli Travertine 0.27% − 16.67% [31] Wackestone 0.05% − 0.25% - GS-GSN-GSP 13.00% − 45.00% [34, 39, 48] OT1-OT2-OT3-OT4 33.50% − 44.30% producer data [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Absorption function and its derivative for different choices of parameters. The primary objective of this study is to determine the model parameters in (5) for the materials described in Section 2 by comparing numerical simulations and experiments. To this end, we define the overall quantity of fluid absorbed by the specimen at time t as (6) Q(t) = ρl Z H 0 θ(z, t) dz, which represents a physically measura… view at source ↗
Figure 4
Figure 4. Figure 4: Experimental convergence and performances of the FTCS and MOL methods. 4. Experimental Data Analysis As discussed earlier, the total amount of fluid absorbed by the specimen, defined in (6), represents an experimentally measurable quantity that establishes a connection between the mathematical model and physical experiments. Because of its intrinsic physical inter￾pretation, Q(t) is expected to exhibit a m… view at source ↗
Figure 5
Figure 5. Figure 5: Monotonicity-preserving data differential reconstruction pro￾cedure applied to the averaged Carrara marble (left) and travertine (right) datasets. 0 100 200 300 400 500 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Q(t) [g/m 2 ] Averaged Wackestone Data Original Data Reconstructed Data 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 10-3 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.2… view at source ↗
Figure 6
Figure 6. Figure 6: Monotonicity-preserving data differential reconstruction pro￾cedure applied to the averaged wackestone (left) and GS (right) datasets. • n0 ∈ N , the porosity of the material, accounting for the ranges in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Monotonicity-preserving data differential reconstruction pro￾cedure applied to the averaged GSN (left) and GSP (right) datasets. solution to (19) p ∗ = arg min p∈Ω  ℓ(Qdata, Qnum(p; ∆z, ∆t)) + φ(Q Ndata data , QkNdata (p; ∆z, ∆t)  , where the loss function ℓ(·) measures the discrepancy between the observed and simulated dynamics throughout the entire time interval [0,T] and φ(·) represents the final cost… view at source ↗
Figure 8
Figure 8. Figure 8: Simulation of the data-informed mathematical model, com￾pared against the reconstructed experimental imbibition curve for Carrara marble (left) and travertine (right). 6. Conclusions and future perspectives This work presents a comprehensive, data-informed mathematical framework specifi￾cally designed to characterize capillary water absorption in porous materials commonly employed in contexts of relevance … view at source ↗
Figure 9
Figure 9. Figure 9: Simulation of the data-informed mathematical model, com￾pared against the reconstructed experimental imbibition curves for Tivoli travertine with parallel (left) and perpendicular (right) anisotropy planes. 0 100 200 300 400 500 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Q(t) [g/m 2 ] Wackestone Reconstructed Data PDE Model Outcome 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25… view at source ↗
Figure 10
Figure 10. Figure 10: Simulation of the data-informed mathematical model, com￾pared against the reconstructed experimental imbibition curves for wacke￾stone (left) and GS (right). noise and instrumental limitations, while retaining the physical coherence of the underly￾ing monotone water uptake process. A dedicated calibration procedure, based on particle swarm optimization and enhanced by a multigrid hierarchical strategy, is… view at source ↗
Figure 11
Figure 11. Figure 11: Simulation of the data-informed mathematical model, com￾pared against the reconstructed experimental imbibition curves for GSN (left) and GSP (right). tool for analyzing water absorption phenomena in porous geomaterials, with potential applications in cultural heritage preservation. Given the promising results of this study, several potential directions for future work emerge. A natural extension would in… view at source ↗
Figure 12
Figure 12. Figure 12: Simulation of the data-informed mathematical model, com￾pared against the experimental imbibition curves for different kinds of mortars. [2] R. Askey. The 1839 Paper on Permutations: Its Relation to the Rodrigues Formula and Further Developments. In Simon L. Altmann and Eduardo L. Ortiz, editors, Mathematics and Social Utopias in France: Olinde Rodrigues and His Times, volume 28 of History of Mathematics,… view at source ↗
read the original abstract

In this work, we characterize the water absorption properties of selected porous materials through a combined approach that integrates laboratory experiments and mathematical modeling. Specifically, experimental data from imbibition tests on marble, travertine, wackestone and mortar mock-ups are used to inform and validate the mathematical and simulation frameworks. First, a monotonicity-preserving fitting procedure is developed to preprocess the measurements, aiming to reduce noise and mitigate instrumental errors. The imbibition process is then simulated through a partial differential equation model, with parameters calibrated against rough and smoothed data. The proposed procedure appears particularly effective to characterize absorption properties of different materials and it represents a reliable tool for the study and preservation of cultural heritage.

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 develops a pipeline to characterize water absorption in porous materials (marble, travertine, wackestone, mortar) by combining imbibition experiments with a monotonicity-preserving smoother for data preprocessing and a PDE model whose parameters are calibrated to both raw and smoothed curves. The authors conclude that the combined procedure is particularly effective for material characterization and a reliable tool for cultural-heritage studies.

Significance. If the validation gaps are closed, the integration of monotonicity-preserving fitting with PDE calibration could supply a reproducible workflow for non-destructive assessment of porous heritage materials. The explicit use of both rough and smoothed data for calibration is a constructive design choice that, once quantified, would strengthen the claim of reliability.

major comments (2)
  1. [Abstract] Abstract: the assertion that the procedure 'appears particularly effective' rests on the unverified premise that the monotonicity-preserving smoother leaves the physically relevant absorption rate unchanged; no table or figure compares the resulting PDE permeability/porosity parameters (or their confidence intervals) obtained from raw versus smoothed data.
  2. [Results / Calibration section] The manuscript supplies no cross-validation metrics, error bars on fitted coefficients, or stability tests with respect to the smoothing parameter; without these the central claim that the pipeline is reliable for heritage applications cannot be assessed quantitatively.
minor comments (2)
  1. The abstract would be clearer if it stated the number of replicate imbibition tests performed per material and the precise functional form of the PDE (including boundary conditions).
  2. Notation for the smoothing parameter and the calibrated constants should be introduced consistently in the text and any accompanying equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important opportunities to strengthen the quantitative support for our claims. We agree that direct comparisons and additional validation metrics are needed to substantiate the reliability of the pipeline for heritage applications. We respond to each major comment below and will incorporate the suggested improvements in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the procedure 'appears particularly effective' rests on the unverified premise that the monotonicity-preserving smoother leaves the physically relevant absorption rate unchanged; no table or figure compares the resulting PDE permeability/porosity parameters (or their confidence intervals) obtained from raw versus smoothed data.

    Authors: We acknowledge that the current manuscript does not include an explicit side-by-side comparison of the calibrated PDE parameters obtained from the raw versus smoothed data. In the revised version we will add a dedicated table (and, if space permits, a supplementary figure) reporting the fitted permeability and porosity values together with their confidence intervals for both datasets across all four materials. This addition will directly address whether the monotonicity-preserving smoother preserves the physically relevant absorption characteristics and will allow readers to evaluate any systematic differences introduced by preprocessing. revision: yes

  2. Referee: [Results / Calibration section] The manuscript supplies no cross-validation metrics, error bars on fitted coefficients, or stability tests with respect to the smoothing parameter; without these the central claim that the pipeline is reliable for heritage applications cannot be assessed quantitatively.

    Authors: We agree that quantitative validation is essential for the reliability claim. In the revision we will add error bars on the fitted coefficients based on the calibration procedure and will include stability tests that vary the smoothing parameter over a plausible range, reporting the resulting variation in the PDE parameters. Full k-fold cross-validation is limited by the small number of experimental replicates per material; we will therefore provide leave-one-out validation results where feasible and, if additional independent measurements become available, compare them against the calibrated model predictions. These changes will supply the quantitative assessment requested. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained against experimental inputs

full rationale

The paper develops a monotonicity-preserving fitting procedure as a preprocessing step to reduce noise in imbibition test data from porous materials, then calibrates a PDE model against both the raw measurements and the smoothed versions. This constitutes a standard data-informed workflow in which the fitting serves as an independent noise-mitigation tool rather than a self-referential definition of the target absorption properties. No step reduces by construction to its own outputs (e.g., no fitted parameter is relabeled as an independent prediction, and no uniqueness theorem or self-citation is invoked to force the modeling choices). The central characterization therefore rests on external laboratory data and remains falsifiable against independent physical measurements, yielding a self-contained derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a standard capillary-flow PDE plus a monotonicity constraint on the data smoother is sufficient to extract material-specific absorption parameters. No new physical entities are introduced.

free parameters (1)
  • permeability and porosity-related constants in the PDE
    These are calibrated to the experimental mass-gain curves; their values are not derived from first principles.
axioms (1)
  • domain assumption The imbibition process can be adequately described by a deterministic PDE with constant material coefficients.
    Invoked when the authors state that the imbibition process is simulated through a partial differential equation model.

pith-pipeline@v0.9.0 · 5662 in / 1456 out tokens · 27749 ms · 2026-05-19T10:53:20.492965+00:00 · methodology

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