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A tensor completion method trained at only three temperatures extrapolates diffusion coefficients across a 110 K span, roughly halving the error of the established semi-empirical model.

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2026-08-02 20:26 UTC pith:H2UZ32IA

load-bearing objection A genuinely useful engineering extension—tensor completion for temperature-dependent D∞ with new NMR data—but the temperature extrapolation is a post-hoc linear fit that needs a per-system baseline before the 'astonishing' claim earns its keep. the 4 major comments →

arxiv 2602.23142 v1 pith:H2UZ32IA submitted 2026-02-26 cs.LG

Prediction of Diffusion Coefficients in Mixtures with Tensor Completion

Zeno Romero , Kerstin M\"unnemann , Hans Hasse , Fabian Jirasek This is my paper
classification cs.LG
keywords diffusion coefficientstensor completionTucker decompositionmatrix completionactive learningtemperature extrapolationinfinite dilutionBayesian inference
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 introduces a tensor completion method (TCM) for predicting infinite-dilution diffusion coefficients in binary mixtures over a range of temperatures. The model is trained on experimental data at just three temperatures (298, 313, and 333 K) and then learns to extrapolate: the two latent temperature features come out linear in T, so predictions at any temperature between 268 and 378 K follow from a straight-line fit. The paper claims that across 98 independent data points in this 110 K range the TCM achieves a relative mean absolute error of 0.118, less than half the 0.263 error of the established SEGWE semi-empirical model, and that it also beats isothermal matrix completion models at each of the three training temperatures. A reader should care because reliable temperature-dependent diffusion data are scarce, and this result suggests a joint tensor model can replace a family of single-temperature models while gaining accuracy.

Core claim

The central claim is that a low-rank Tucker decomposition of a solute × solvent × temperature tensor, trained jointly at 298, 313, and 333 K, learns temperature features wγ(T) that are linear in T. Fitting a line wγ(T)=Aγ+BγT to those three learned values turns the discrete tensor model into a continuous-temperature predictor. With that linear extrapolation, the TCM reaches a relative mean absolute error of 0.118 on 98 literature data points between 268 and 378 K, versus 0.263 for the SEGWE model; at the three training temperatures, leave-one-out analysis shows lower errors than both SEGWE and temperature-specific matrix completion models. The paper also reports that incorporating 19 new sys

What carries the argument

The central object is a third-order tensor with dimensions solutes, solvents, and temperature, modeled by a Tucker decomposition: ln D∞ij(T) = Σ uiα vjβ wγ(T) καβγ, where u and v are temperature-independent solute/solvent features, w are temperature features, and κ is the core tensor coupling them. The load-bearing mechanism is the temperature feature vector wγ(T): after training, the discrete wγ values are fitted to a straight line wγ(T)=Aγ+BγT, converting the discrete tensor into a continuous-temperature predictor. The Bayesian two-step training uses SEGWE predictions as priors and a Cauchy likelihood on experimental data, yielding posterior uncertainties that drive both evaluation and act

Load-bearing premise

The load-bearing premise is that the two learned temperature features vary linearly with temperature, so a straight line fitted through only three training points (298, 313, 333 K) remains valid across the much larger extrapolation range of 268-378 K.

What would settle it

Measure infinite-dilution diffusion coefficients for a set of binary systems with strongly non-Arrhenius temperature behavior (for example, water at temperatures near 268 K or systems approaching a critical point) between 268 and 378 K; if the TCM's linear-in-temperature predictions deviate systematically beyond its reported ~12% error, the linear wγ(T) assumption fails.

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

If this is right

  • If the claim is correct, isothermal matrix completion models are no longer needed for infinite-dilution diffusion coefficients: a single tensor model trained jointly across temperatures improves accuracy at the training temperatures and simultaneously supplies temperature extrapolation.
  • The linear-temperature features imply that, within 268-378 K, the log diffusion coefficient for the studied mixtures is effectively linear in temperature, so interpolation between measured temperatures adds no systematic error beyond the model's stated ~12% rMAE.
  • The active-learning loop can guide scarce experimental resources: 19 targeted NMR measurements at three temperatures each produced measurable accuracy gains even though they filled only 1.8% of the tensor.
  • The method provides prediction uncertainties from the Bayesian posterior, which can be used directly to flag which unmeasured solute-solvent pairs carry the largest residual risk.
  • The 98 data points used for the continuous-temperature evaluation lie entirely outside the three training temperatures, making the reported result a genuine extrapolation test rather than an interpolation artifact.

Where Pith is reading between the lines

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

  • The 110 K extrapolation rests on a linearity assumption that is fitted to only three points; the paper's own caution that accuracy worsens with temperature above the training range suggests that a physically motivated scaling (e.g., Arrhenius or a viscosity-corrected form) might extend the valid range and is worth testing against the current linear fit.
  • A direct test of the mechanism would be to replace the linear fit in T with other two-parameter temperature scalings and compare leave-one-out errors on the same 98 independent points, isolating whether the reported advantage comes from the tensor structure or from the particular extrapolation curve.
  • One could extend the active-learning procedure to query temperatures as well as mixtures, since the current AL loop is intentionally restricted to the three discrete temperatures; a continuous-temperature query strategy might reveal whether the linear wγ(T) assumption holds where new data are most informative.
  • If the method transfers, it should be possible to use the same TCM to estimate mutual diffusion coefficients at finite concentration via the Vignes extrapolation, turning a temperature-dependent infinite-dilution prediction into a wider engineering tool.

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

4 major / 5 minor

Summary. The paper proposes a hybrid tensor completion method (TCM) for predicting infinite-dilution diffusion coefficients in binary mixtures as a function of temperature. The TCM uses a Tucker decomposition with SEGWE-based priors, is trained at 298 K, 313 K, and 333 K, and is then extended to continuous temperatures by fitting the learned temperature features linearly in T. The authors evaluate the TCM on 98 DDB data points at 268–378 K, report rMAE 0.118 versus 0.263 for SEGWE, and also perform active learning experiments with 19 new PFG-NMR measurements. The central claims are that the TCM outperforms isothermal matrix completion methods and SEGWE at the three training temperatures, and that it extrapolates accurately over the wider temperature range.

Significance. If the claims hold, the TCM would be a practically useful method for temperature-dependent prediction of a scarce transport property, and the combination of tensor completion with active learning is a sensible direction. The manuscript includes several strengths: the 98-point DDB holdout at temperatures outside the training set is a genuine evaluation, the authors provide Stan code and some trained parameters, and the AL measurements are reported with uncertainties. However, the key scientific novelty — that the model 'learned' a temperature extrapolation — is weakened by the fact that Eq. (8) imposes linearity of ln D in T after the fact, using only three temperatures. The comparison lacks a per-system linear or Arrhenius baseline, so the reported extrapolation accuracy is not yet evidence that the tensor structure adds value beyond a simple empirical temperature scaling. The paper is therefore publishable in principle, but the central claim needs to be reframed and supported by additional baselines and a nested model-selection evaluation.

major comments (4)
  1. [§Analysis of the Temperature Features, Eq. (8) and Table 2] The continuous-temperature capability rests entirely on wγ(T)=Aγ+BγT fitted to the three learned wγ values. Since the TCM uses rw=2 and u,v,κ are temperature-independent, this makes ln D_ij(T) = C_ij + D_ij·T exactly for every pair (i,j). A linear regression to three points with R²>0.99 is not independent evidence of linearity; with only three points and two parameters, the fit is essentially an interpolation. The 98-point DDB holdout is real, but the paper never compares against the obvious baseline: fitting a per-system linear (or Arrhenius, ln D vs 1/T) model using only that system's available 298/313/333 K data and evaluating on the same 98 points. If that baseline achieves rMAE comparable to 0.118, the 'most astonishing' extrapolation claim reduces to the physical near-linearity of ln D over this range rather than a learned capability of the tensor model. This comparison is essentia
  2. [§Computational Details and Evaluation, Figure S2] Hyperparameters r_u, r_v, r_w are selected by leave-one-out cross-validation on the same experimental dataset that is later used to report the final LOO errors. This is a model-selection-on-the-test-set problem and can make the reported rMAE/rMSE optimistic. The authors should use nested cross-validation or at least report the model-selection bias, e.g., the difference between the selected-configuration LOO error and the error of the configuration that would be chosen by an independent validation split. Additionally, rMAE and rMSE are reported as point estimates without uncertainty; given the small number of test points in some temperature bins, bootstrap confidence intervals would be helpful.
  3. [§Results and Discussion, Figure 7] The continuous-temperature test set contains only solute–solvent pairs that already appear in the training matrix; the model is evaluated at unseen temperatures for known components, not for unseen pairs. The abstract and conclusion claim prediction for 'binary mixtures' and 'any temperature' in a broad range, which suggests generalization to new mixtures as well. That claim is currently unverified: the TCM's matrix-completion capability for unseen pairs is demonstrated only at 298, 313, and 333 K via LOO, and the continuous-temperature extrapolation is shown only for existing pairs. The manuscript should clearly state this limitation or evaluate extrapolation on a subset of truly unseen pairs at out-of-sample temperatures.
  4. [§Analysis of the Temperature Features] The statement that the TCM 'learned' the temperature correlation without any physical information about temperature is overstated. The first training step is performed on the complete synthetic SEGWE tensor, whose values at 298, 313, and 333 K already encode SEGWE's temperature-dependent solvent viscosities. Thus the temperature features are influenced by a physical model that is itself temperature-dependent. This does not invalidate the empirical holdout, but it weakens the claim that the linear scaling emerges purely from experimental data. The authors should either temper this language or test a version without the SEGWE prior to isolate the contribution of the experimental data.
minor comments (5)
  1. [§Computational Details and Evaluation] The evaluation range is stated as '[268 K, 453 K]' in the Computational Details, while the abstract and Results/Figure 7 restrict the claim to 268–378 K. This inconsistency should be resolved, and the higher-temperature behavior should be discussed explicitly if the model is evaluated up to 453 K.
  2. [Eq. (2)] The noise term changes from εij in Eq. (1) to εijk in Eq. (2), but the tensor index convention is not defined. Use consistent subscripts throughout.
  3. [Figures 1 and 2] The rendered figure captions and axis labels contain garbled token sequences (e.g., '/uni00000014...') in the provided manuscript. This is likely a production/rendering issue, but the figures should be clean and legible in the final version.
  4. [Table 2] The regression statistics in Table 2 are reported for two fitted lines each based on only three points. Reporting R² and MSE to four significant figures gives a false impression of precision. A plot with the three points and the fit, plus residual values, would be more informative.
  5. [§Improvement of TCM by Active Learning] The AL improvement is reported as rMSE changes from 0.18 to 0.15, 0.10 to 0.08, and 0.07 to 0.06 at the three temperatures. These are small differences and likely within the noise of the LOO evaluation; Figure S5 visually supports only a modest effect. The text should avoid overemphasizing 'substantial improvements' without uncertainty quantification or significance testing.

Circularity Check

0 steps flagged

No significant circularity: the 110 K temperature extrapolation is a genuine out-of-sample test; only the post-hoc 'learned linearity' framing warrants a mild caveat.

full rationale

The tensor-completion derivation is not circular. The continuous-temperature claim rests on Eq. (8), w_gamma(T) = A_gamma + B_gamma T, with A_gamma/B_gamma fitted to the model's own latent temperature features at the three training temperatures (298/313/333 K); the 98 DDB test points at 268-378 K are excluded from training ("No data for these temperatures were, however, used for training the TCM"), so the reported rMAE = 0.118 holdout evaluation genuinely tests the extrapolation rather than reproducing a fit. The "most astonishing ... learned a correlation ... purely from experimental data" framing is post hoc: a linear fit through three points with R^2 > 0.99 is not independent evidence of linearity, and the linear law is imposed after the three w_gamma values were observed. But this is an overstatement of an empirically confirmed assumption, not a reduction of the prediction to its inputs. The TCM-vs-MCM and TCM-vs-SEGWE comparisons are held-out-system evaluations (leave-one-out over solute-solvent pairs; 98-point continuous test set), so the SEGWE prior does not force the outcome. Self-citations ([9] MCM-Whisky/hybrid Bayesian training, [31] active-learning query strategy, [37] tensor completion for activity coefficients) supply methodology that is re-described inline with explicit equations, and the central claims are benchmarked against external DDB data and the published SEGWE model, so no load-bearing step reduces to a self-citation. The paper itself acknowledges the limitation ("the linear scaling with T should be used only within the specified 268 K to 378 K range"). Remaining scientific-credit concerns (no per-system linear-extrapolation baseline, so part of the gain may reflect physical near-linearity of ln D in T; test pairs lie inside the training matrix) are not circularity. Score 2 reflects the post-hoc linearity framing as the only mild caveat.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The model leans on several external inputs: licensed DDB/literature data, SEGWE priors with DIPPR viscosities, and a post-hoc linear temperature-feature fit. No new physical entities are introduced; the main load-bearing free parameters are the four linear coefficients that enable temperature extrapolation.

free parameters (4)
  • Linear temperature-feature coefficients A1, B1, A2, B2 = A1=1.195, B1=-6.179e-3; A2=8.271, B2=-2.648e-2
    Fit to the two latent temperature features wγ at only 298, 313, and 333 K; these four coefficients carry the entire continuous-temperature extrapolation.
  • Latent feature dimensions ru=rv=rw=2 = 2
    Hyperparameters selected by leave-one-out CV on the experimental data used for evaluation; the paper notes rw=4 gave slightly lower rMAE (0.168) but rw=2 was chosen to keep parameter count minimal.
  • Prior-scale factor σbar=0.5 and Cauchy scale λ=0.2 = 0.5, 0.2
    Hand-set Bayesian prior hyperparameters adopted from previous MCM work; they control how strongly SEGWE priors influence final predictions.
  • SEGWE effective density ρeff = 627 kg/m3
    Adopted from the SEGWE literature and used to generate the synthetic prior data; not fitted in this paper but load-bearing for the prior.
axioms (6)
  • domain assumption The literature and DDB values for D∞ij, including those obtained by extrapolation from finite concentrations, are accurate representations of infinite-dilution diffusion coefficients.
    All training and holdout comparisons inherit the accuracy of externally reported data; the paper applies the Großmann et al. extrapolation scheme without independent verification.
  • domain assumption SEGWE predictions are useful priors for diffusion coefficients across solutes, solvents, and temperatures.
    The hybrid Bayesian training step relies on SEGWE synthetic data; if SEGWE were systematically biased in a way that experimental data cannot override, predictions would inherit that bias.
  • domain assumption A rank-2 Tucker decomposition captures the structure of ln D∞ij across solutes, solvents, and temperatures.
    The model form assumes the log-diffusion tensor is low-rank; the paper justifies this only by LOO performance, not by physical argument.
  • ad hoc to paper The temperature features wγ(T) are linear in T over 268-378 K.
    Eq. (8) is fitted to wγ at three temperatures; it is the key assumption enabling extrapolation and is not derived from Stokes-Einstein or SEGWE.
  • domain assumption Infinite-dilution values can be obtained by linear extrapolation of PFG NMR self-diffusion coefficients measured at 0.005, 0.01, and 0.025 mol/mol.
    The 19 new D∞ij values assume negligible curvature in D vs concentration over this range.
  • domain assumption Test data at 268-378 K from the DDB are comparable in quality and composition to the training data at 298-333 K.
    The continuous-temperature evaluation treats these 98 points as ground truth for systems whose components largely overlap the training matrix.

pith-pipeline@v1.3.0-alltime-deepseek · 28180 in / 14565 out tokens · 129110 ms · 2026-08-02T20:26:22.116807+00:00 · methodology

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read the original abstract

Predicting diffusion coefficients in mixtures is crucial for many applications, as experimental data remain scarce, and machine learning (ML) offers promising alternatives to established semi-empirical models. Among ML models, matrix completion methods (MCMs) have proven effective in predicting thermophysical properties, including diffusion coefficients in binary mixtures. However, MCMs are restricted to single-temperature predictions, and their accuracy depends strongly on the availability of high-quality experimental data for each temperature of interest. In this work, we address this challenge by presenting a hybrid tensor completion method (TCM) for predicting temperature-dependent diffusion coefficients at infinite dilution in binary mixtures. The TCM employs a Tucker decomposition and is jointly trained on experimental data for diffusion coefficients at infinite dilution in binary systems at 298 K, 313 K, and 333 K. Predictions from the semi-empirical SEGWE model serve as prior knowledge within a Bayesian training framework. The TCM then extrapolates linearly to any temperature between 268 K and 378 K, achieving markedly improved prediction accuracy compared to established models across all studied temperatures. To further enhance predictive performance, the experimental database was expanded using active learning (AL) strategies for targeted acquisition of new diffusion data by pulsed-field gradient (PFG) NMR measurements. Diffusion coefficients at infinite dilution in 19 solute + solvent systems were measured at 298 K, 313 K, and 333 K. Incorporating these results yields a substantial improvement in the TCM's predictive accuracy. These findings highlight the potential of combining data-efficient ML methods with adaptive experimentation to advance predictive modeling of transport properties.

Figures

Figures reproduced from arXiv: 2602.23142 by Fabian Jirasek, Hans Hasse, Kerstin M\"unnemann, Zeno Romero.

Figure 1
Figure 1. Figure 1: Experimental data for liquid-phase diffusion coeffic [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experimental data for liquid-phase diffusion coeffic [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the hybrid TCM for pred [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Active learning workflow for the targeted improvemen [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boxplot of the AREij of the predicted D∞ ij with SEGWE,2 MCM,9 and the developed TCM. MCM and TCM results were obtained using leave-one-out analysis, the SEGWE model was used as proposed by the original authors.2 Boxes represent interquartile ranges (IQR) and whiskers represent 1.5 IQR [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the latent temperature features wγ (or rather their means) calculated by the TCM trained on the entire discrete data set as a function of T. 300 310 320 330 T / K 0.8 0.6 0.4 0.2 0.0 0.2 0.4 w w1 (discrete) w2 (discrete) w1 (continuous) w2 (continuous) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: γ Aγ Bγ R2 MSE 1 1.195 -6.179 ·10−3 0.9939 4.81 ·10−5 2 8.271 -2.648 ·10−2 0.9997 3.18 ·10−5 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Boxplot of the AREij of the predicted D∞ ij with SEGWE2 and the TCM as a function of T. The numbers above each box indicate the number of data points per bin, horizontal lines represent the median, boxes represent the IQR, and whiskers represent 1.5 IQR. The TCM maintains high performance across the temperature range from 268 to 378 K, with a total rMAE of 0.118, while the SEGWE model has a total rMAE of 0… view at source ↗

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Forward citations

Cited by 1 Pith paper

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    A matrix-completion model hybridized with the Bromley equation predicts activity coefficients and osmotic coefficients for 9,296 aqueous electrolytes at 298 K, including unmeasured ones.

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages · cited by 1 Pith paper

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    Release. 20 24; https://zenodo.org/ doi/10.5281/zenodo.13848108. (7) Evans, R.; Poggetto, G. D.; Nilsson, M.; Morris, G. A. Impro ving the interpretation of small molecule diffusion coefficients. Analytical Chemistry 2018, 90, 3987–3994. S11

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    (3) Romero, Z.; M¨ unnemann, K.; Hasse, H.; Jirasek, F

    https://www.ddbst.c om (accessed 2025-07- 01). (3) Romero, Z.; M¨ unnemann, K.; Hasse, H.; Jirasek, F. Improveme nt of diffusion coefficient prediction by active learning. Journal of Physical Chemistry B 2025, –, –. (4) Mross, S.; Schmitt, S.; Stephan, S.; M¨ unnemann, K.; Hass e, H. Diffusion coefficients in mixtures of poly(oxymethylene) dimethyl ethers with ...

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