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REVIEW 3 major objections 4 minor 62 references

Kolmogorov-Arnold networks, with spline-based learnable activations, can replace costly Gibbs-energy-minimization solvers for geochemical equilibria, cutting errors versus multilayer perceptrons by 62% and evaluation time by up to 93%.

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2026-08-04 05:45 UTC pith:55HACMNP

load-bearing objection A credible, well-documented first KAN application to radionuclide coprecipitation surrogates, but the headline KAN-over-MLP claim rests on a single benchmark and single training runs. the 3 major comments →

arxiv 2603.15307 v1 pith:55HACMNP submitted 2026-03-16 cs.LG physics.chem-ph

A Kolmogorov-Arnold Surrogate Model for Chemical Equilibria: Application to Solid Solutions

Leonardo Boledi , Dirk Bosbach , Jenna Poonoosamy This is my paper
classification cs.LG physics.chem-ph MSC 68T0765D07
keywords Kolmogorov-Arnold networkschemical equilibrium surrogatessolid solutionsradium coprecipitationreactive transportgeochemical modelingspline activationscement hydration
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that Kolmogorov-Arnold networks, whose learnable spline functions replace fixed activations, are a practical surrogate for chemical equilibrium calculations. On a cement-hydration benchmark, the paper reports a 62% reduction in absolute error and 59% in relative error compared with multilayer perceptrons at similar parameter counts. On radium coprecipitation into sulfate solid solutions, from mechanical mixtures through non-ideal binary and ternary solid solutions, it reports median prediction errors near 1e-3 and no test predictions above a 10% error threshold, together with 87–93% reductions in evaluation time. If correct, this matters because reactive transport simulations call geochemical solvers billions of times, so accurate fast surrogates could enable large uncertainty and safety analyses for nuclear waste disposal.

Core claim

The paper's central claim is that for the chemical systems and input ranges studied, Kolmogorov-Arnold networks are strictly better surrogates than multilayer perceptrons: lower absolute and relative errors at equal or smaller parameter counts, no degradation as thermodynamic complexity increases to non-ideal ternary solid solutions, and evaluation times roughly an order of magnitude faster than the reference Gibbs-energy-minimization solver. The KAN replaces each fixed neuron activation with a learnable B-spline, letting the network fit sharp composition-dependent solubility behavior with fewer parameters. The study is, per the authors, the first surrogate model for radionuclide coprecipita

What carries the argument

The Kolmogorov-Arnold network is the load-bearing object: its learnable univariate B-spline activation functions sit on every connection, and the composition of these one-dimensional functions plus addition is what approximates the equilibrium state mapping. The paper also relies on a regular solid-solution model with binary interaction parameters to generate training data of increasing thermodynamic complexity, and on low-discrepancy Sobol sampling to cover the input space.

Load-bearing premise

That the Gibbs-energy-minimization solver with its thermodynamic database gives the correct equilibrium, and that accuracy measured on one-time Sobol-sampled equilibrium states carries over to the time-stepped reactive transport settings where the surrogate would be deployed.

What would settle it

Generate equilibria with the reference solver at compositions and temperatures outside the sampled ranges, especially near solid-solution miscibility or phase-boundary transitions, and check whether KAN errors stay below the 10% threshold; alternatively, couple the KAN to a reactive transport code and track total mass balance over hundreds of time steps. If either test shows errors above 10% or systematic mass drift, the central claim that KANs can replace the solver fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • On the tested input ranges, the KAN surrogate can replace the Gibbs-energy-minimization solver for one-shot equilibrium predictions, keeping median relative errors near 0.1%.
  • The 87–93% per-evaluation speedup translates, in principle, to reactive transport simulations that currently spend most of their runtime on chemistry.
  • KANs achieve this with fewer trainable parameters than MLPs, lowering memory requirements for storing the model inside large simulation codes.
  • Treating temperature as an input variable opens the door to temperature-scanning safety assessments that were previously too expensive.
  • The longer KAN training time, on the order of minutes, is a one-time cost amortized over billions of subsequent evaluations.

Where Pith is reading between the lines

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

  • The reported speedups are per-equilibrium-call, not end-to-end reactive transport speedups; time-dependent coupling, mass-conservation drift, and error accumulation remain unmeasured, so the practical simulation speedup could be smaller.
  • Because the network is trained entirely on solver-generated points, its accuracy is bounded by the thermodynamic database; a systematically wrong reference model would make the surrogate confidently wrong.
  • Error statistics over Sobol samples may not hold near phase boundaries or outside the sampled ranges; extrapolation testing would tell whether the surrogate can be trusted for exploratory safety assessments.
  • The spline activations could be inspected to recover interpretable composition-to-solubility relationships, offering a route from black-box prediction to thermodynamic insight.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes Kolmogorov-Arnold networks (KANs) as surrogate models for geochemical equilibrium calculations. It first benchmarks KANs against a published multilayer perceptron (MLP) on a CaO-SiO2-H2O cement hydration dataset, reporting 62% and 59% reductions in absolute and relative error, respectively. It then builds GEM-Selektor datasets for three radium sulfate cases of increasing thermodynamic complexity: a mechanical mixture, a binary (Ba,Ra)SO4 solid solution, and a ternary (Sr,Ba,Ra)SO4 solid solution. For these cases, the paper reports median relative errors near 1e-3, no test predictions exceeding a 10% error threshold for the binary and ternary solid-solution cases, and 87-93% reductions in evaluation time relative to GEM-Selektor. The datasets, GEM-Selektor input files, and generation scripts are made publicly available.

Significance. If the reported results are robust, the paper would provide useful evidence that KANs are viable surrogates for expensive Gibbs-energy-minimization solvers over bounded input ranges, and it would be the first surrogate-model study of radionuclide co-precipitation in sulfate solid solutions. The open-data policy and the use of a well-established thermodynamic database are strengths, as is the systematic progression from mechanical mixtures to binary and ternary solid solutions. The central quantitative claim, however, is the KAN-over-MLP improvement on the cement benchmark, and that claim rests on a comparison whose fairness and statistical reliability are not yet established. The work is a solid application-oriented contribution if the comparison is placed on equal footing and uncertainty quantification is added.

major comments (3)
  1. [Section 4.1, Tables 2-3] The headline 62%/59% improvements compare two KAN configurations against an MLP taken verbatim from Prasianakis et al. with no Optuna re-tuning on the current train/validation/test split. Sections 3.1 and 3.2 state that both MLPs and KANs use Optuna for hyperparameter selection, but in Section 4.1 only the KANs appear to be selected or tuned for this dataset; the MLP is fixed as the 'reference model.' This asymmetry is load-bearing because the central claim is architectural superiority. Please retune the MLP on the same split with a comparable search budget, or clearly justify why the literature MLP is the correct baseline, and report the resulting error distributions.
  2. [Section 4.2, dataset split] The text says datasets of size 2^m are 'split in half to obtain the training and validation sets. From the latter, 5000 points are subtracted for testing.' If the test points are a subset of the validation set used for early stopping or learning-rate scheduling, the test errors are optimistically biased. Please clarify whether the test set was held out before training and never used for model selection; if not, redesign the split so that training, validation, and test subsets are disjoint.
  3. [Figures 4-7, Tables 2-5] All reported errors, medians, high-error counts, and run-time improvements come from single training runs with no seed-level statistics. Given the noise in neural-network training, a single favorable seed cannot be distinguished from genuine architecture-level advantage, especially for the claim that 'no predictions exceed the 10% error threshold' in Sections 4.3-4.4. Please repeat training for at least 5-10 seeds, report median/interquartile ranges or worst-case error, and state whether the no-threshold-exceedance claim holds across all seeds.
minor comments (4)
  1. [Equation (6)] The nested composition formula for the MLP has malformed superscripts and parentheses (e.g., 'f n+1' and misplaced b terms). Please rewrite it with consistent indexing for readability.
  2. [Tables 2-3] The variable 'AmorfSi' appears to be a typo; please use a consistent naming convention. Also, define the units for the RMSE columns or note that they are in the scaled/log-transformed space.
  3. [Figure 4-7] The box plots lack axis labels and legend definitions. Please add explicit labels (e.g., 'Output variable' on the x-axis, 'Relative error' on the y-axis) and explain what the red counts represent in captions.
  4. [Section 4.5 and 5] The discussion appropriately acknowledges that only one-time equilibria were studied and that mass conservation remains an issue for time-dependent reactive transport. Please make the scope limitation more explicit in the abstract or introduction so that the strong wording in the Conclusion does not overstate readiness for RTM deployment.

Circularity Check

0 steps flagged

No circular derivation: KAN/MLP accuracy claims are empirical supervised fits against external solver outputs; thermodynamic parameters are imported literature inputs, not fitted or derived here.

full rationale

The paper makes no claimed derivation from first principles; its central results are supervised-learning accuracies and runtime measurements. Thermodynamic inputs (Margules parameters, PSI-Nagra database, GEM-Selektor mixing models) are imported from prior external literature (e.g., Vinograd et al. 2018) and are not fitted or derived in this work. The GEM-Selektor-generated datasets serve as training labels, and evaluating on held-out solver points is standard supervised approximation testing, not a circular 'prediction' in the derivational sense. The KAN-versus-MLP comparison is an empirical benchmark: Section 4.2 states both models are trained on identical datasets, and the cement benchmark uses a pre-published reference MLP from Prasianakis et al. Possible asymmetric hyperparameter tuning is a benchmark-fairness and correctness concern, not circularity, because the comparison does not reduce to an identity, fitted parameter, or self-citation. The self-citations to prior geochemical studies provide thermodynamic context and parameter values rather than load-bearing uniqueness arguments or ansatz smuggling. Section 4.5 explicitly acknowledges the scope limitation to one-time equilibrium predictions and the risk of mass-conservation error accumulation in reactive transport, which is an external-validity caveat, not a circular step. Thus no specific reduction to inputs by construction can be exhibited.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities or fitted thermodynamic constants; it imports all chemistry from prior work. What is fitted is the ML model itself (hyperparameters chosen by Optuna and hand-set training settings). The key unstated premise is that GEM-Selektor's thermodynamic output is the correct target for the surrogate.

free parameters (4)
  • KAN architecture hyperparameters, cement KAN1 = 4 hidden layers, 28 neurons/layer, spline order 7, 10 grid points
    Selected to match roughly one-third of the MLP parameter count; part of the claimed 62%/59% improvement.
  • KAN architecture hyperparameters, cement KAN2 = 5 hidden layers, 40 neurons/layer, spline order 8, 12 grid points, ~1.5e5 parameters
    Chosen to match the MLP parameter count; used for the headline improvement in Tables 2-3.
  • KAN architecture hyperparameters, radium cases = Mechanical mixing: Optuna-selected up to 4 layers/24 neurons/order 8/15 grid points; binary: 3 layers/24 neurons/order 8
    Selected via Optuna on validation sets; central to the reported low median errors.
  • Training hyperparameters (batch size, learning rate, scheduler, epochs) = Batch size 192, initial learning rate 0.01, reduce-on-plateau scheduler, 100-200 epochs
    Chosen by hand; affect all reported results.
axioms (5)
  • domain assumption GEM-Selektor/TSolMod Gibbs-energy minimization is the correct ground truth for chemical equilibria.
    All training and test labels come from this solver (Section 2.2).
  • domain assumption The PSI-Nagra 12/07 thermodynamic database and SUPCRT98/92 equations of state are accurate for the studied species.
    Imported from prior literature in Section 2.2; not validated in this paper.
  • domain assumption The regular solution model with Margules parameters w_RaBa = 2470, w_SrRa = 1750, w_SrBa = 750 J/mol from [47] describes binary and ternary sulfate solid-solution mixing.
    Equations (2)-(4) in Section 2.1 rely on these fitted interaction parameters from prior work.
  • standard math The Kolmogorov-Arnold representation theorem holds for the target functions and the specific spline parameterization is sufficiently expressive.
    Section 3.2, Equations (7)-(8), invoke the theorem as justification for the KAN architecture.
  • domain assumption The cement benchmark dataset published by Prasianakis et al. [23] is valid, and the Sobol-sampled input ranges in Table 1 are representative of repository-relevant conditions.
    Sections 2.2 and 4.1; the chosen ranges are 'inspired by the radium incorporation simulation in [52]'.

pith-pipeline@v1.3.0-alltime-deepseek · 16035 in / 13137 out tokens · 115601 ms · 2026-08-04T05:45:49.063432+00:00 · methodology

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read the original abstract

The computational cost of geochemical solvers is a challenging matter. For reactive transport simulations, where chemical calculations are performed up to billions of times, it is crucial to reduce the total computational time. Existing publications have explored various machine-learning approaches to determine the most effective data-driven surrogate model. In particular, multilayer perceptrons are widely employed due to their ability to recognize nonlinear relationships. In this work, we focus on the recent Kolmogorov-Arnold networks, where learnable spline-based functions replace classical fixed activation functions. This architecture has achieved higher accuracy with fewer trainable parameters and has become increasingly popular for solving partial differential equations. First, we train a surrogate model based on an existing cement system benchmark. Then, we move to an application case for the geological disposal of nuclear waste, i.e., the determination of radionuclide-bearing solids solubilities. To the best of our knowledge, this work is the first to investigate co-precipitation with radionuclide incorporation using data-driven surrogate models, considering increasing levels of thermodynamic complexity from simple mechanical mixtures to non-ideal solid solutions of binary (Ba,Ra)SO$_4$ and ternary (Sr,Ba,Ra)SO$_4$ systems. On the cement benchmark, we demonstrate that the Kolmogorov-Arnold architecture outperforms multilayer perceptrons in both absolute and relative error metrics, reducing them by 62% and 59%, respectively. On the binary and ternary radium solid solution models, Kolmogorov-Arnold networks maintain median prediction errors near $1\times10^{-3}$. This is the first step toward employing surrogate models to speed up reactive transport simulations and optimize the safety assessment of deep geological waste repositories.

Figures

Figures reproduced from arXiv: 2603.15307 by Dirk Bosbach, Jenna Poonoosamy, Leonardo Boledi.

Figure 1
Figure 1. Figure 1: Sketch of an MLP (left) and a KAN (right) with two hidden layers. In KANs, instead of training for weights [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: RMSE on the test set for the cement hydration case. The reference MLP model (blue) is shown against two [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: RRMSE on the test set for the cement hydration case. The reference MLP model (blue) is shown against two [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative error on the test set for the radium uptake case with mechanical mixing. The errors are plotted for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative error on the test set for the radium uptake case with mechanical mixing. The classical MLP model [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relative error on the test set for the radium uptake case with the two-component SS model. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative error on the test set for the radium uptake case with the three-component SS model. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

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