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REVIEW 3 major objections 5 minor 50 references

Trained neural operators can map a tear film's fluorescence intensity time series directly to thickness and osmolarity, replacing the expensive inverse-problem step and opening the door to near-real-time clinical analysis.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 10:56 UTC pith:BK27E3CI

load-bearing objection Genuinely new idea for tear film diagnostics, but the synthetic test split leaks near-duplicates and the experimental validation is not independent; worth a revision, not a desk reject. the 3 major comments →

arxiv 2601.08001 v2 pith:BK27E3CI submitted 2026-01-12 math.NA cs.CVcs.LGcs.NA

Operator learning for models of tear film breakup

Qinying Chen , Arnab Roy , Tobin A. Driscoll This is my paper
classification math.NA cs.CVcs.LGcs.NA MSC 92C35
keywords tear filmdry eye diseasefluorescent imagingscientific machine learningoperator learninginverse problemsneural networkstear film breakup
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.

Tear film breakup is central to dry eye disease, but estimating thickness and osmolarity from fluorescence imaging normally requires solving a computationally costly inverse problem for each case. This paper argues that neural operators trained purely on simulated tear film dynamics can learn the mapping from fluorescence intensity time series directly to thickness and osmolarity, making predictions in a single network evaluation. On synthetic test data from both an ODE and a 1D PDE model, the trained networks achieve roughly one to two digits of accuracy regardless of architecture. On 467 experimental tear breakup traces, they reproduce the reference inverse-solution predictions with discrepancies concentrated in flow- or evaporation-dominated regimes. The authors conclude that the main limit is how much information a single-spot intensity signal carries, not which learning architecture is used.

Core claim

The central claim is that the inverse of the tear film simulation operator—from fluorescence intensity I(t) to thickness h(t) and osmolarity c(t)—can be approximated by a neural network trained on simulated data alone. The paper demonstrates this for two models: a spatially lumped ODE and a one-dimensional radially symmetric PDE that includes evaporation, osmosis, Marangoni flow, and fluorescein transport, with the intensity signal generated by the standard fluorescein relation I = I0 (1 − exp(−φ f h))/(1 + f²). Across three architectures (Fourier feature network, dense PCA network, and dense PCA network augmented with external parameters), the learned map predicts h and c on held-out synthe

What carries the argument

The central object is a learned operator from a 601-point fluorescence intensity time series to the corresponding thickness and osmolarity time series. Training data are produced by Halton-sampling a six-dimensional parameter space for the ODE and PDE models, then filtering out simulations that violate physiological expectations (thickness bounds, regrowth, or increasing intensity). Three network designs—Fourier feature network, dense network on PCA-compressed inputs, and dense PCA network with external parameters appended—encode the intensity signal and decode thickness and osmolarity. This learned map replaces the inverse problem step, and training separate predictors on the two mathematic

Load-bearing premise

The load-bearing premise is that the simulated parameter ranges and rejection filters produce training examples that match the physiologically relevant tear-film behaviors seen in real eyes; if real breakup produces intensity signals outside that distribution, the learned map will be accurate on synthetic tests but wrong clinically, and no independent experimental measurement of thickness or osmolarity tests this premise.

What would settle it

Collect simultaneous fluorescence imaging and an independent thickness measurement (for example, optical coherence tomography or interferometry) on eyes undergoing tear breakup, and compare the network's predicted thickness and osmolarity to those measurements. If the match is systematically poor even for cases that pass the paper's filters, the synthetic-to-clinical transfer fails; a cheaper internal check is to test the trained networks on synthetic parameters deliberately sampled outside the Halton and rejection ranges and look for error blow-up.

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

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If this is right

  • Tear film analysis from fluorescein imaging could run in near-real time during a clinic visit, since each prediction is one network evaluation rather than an iterative optimization.
  • The same operator-learning pipeline can be re-run for any improved forward model, producing a new training set and predictor without redesigning the learning machinery.
  • Differences between ODE-trained and PDE-trained predictors on clinical traces become a quantitative diagnostic for when the simplified ODE misses real tear film physics.
  • The observed one-to-two-digit accuracy on synthetic data quantifies what can be extracted from single-location intensity signals, and points to spatial information as the natural route to better estimates.

Where Pith is reading between the lines

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

  • Editorial extension: because the training distribution is the only source of truth, a clinical deployment should first be validated against independent direct measurements of thickness and osmolarity; the paper's reference values are outputs of the same simulation model, so they cannot validate the underlying physics.
  • Editorial extension: the discrepancy between ODE- and PDE-trained predictors hints that spatial intensity profiles—not just the glob center—carry information the current single-spot inputs discard; training on full 2D spatiotemporal patches is a concrete testable improvement.
  • Editorial extension: the identifiability ceiling the authors observe suggests that physics-constrained predictors, which bring the governing equations back into the network, might push accuracy beyond what unconstrained networks achieve on the same data.

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 / 5 minor

Summary. The manuscript proposes an operator learning framework for tear film breakup. The authors train three neural-network architectures (Fourier feature network, dense network on PCA coefficients, and that with externally known parameters appended) to map fluorescence intensity time series I(t) directly to tear film thickness h(t) and osmolarity c(t), bypassing per-case inverse problem solution. Training data are generated from an ODE model and a 1D PDE model, with parameters drawn from Halton sequences and filtered by ad hoc rejection rules. The networks are evaluated on synthetic test sets and on 467 experimental intensity traces using fits from Driscoll et al. [23] as reference. The paper reports 1–2 digits of relative RMSE on synthetic tests and compares ODE-trained versus PDE-trained predictions on clinical traces.

Significance. If the synthetic generalization results are valid, the paper would demonstrate a practically useful fast surrogate for tear film inverse problems, with potential real-time clinical application. The manuscript is transparent about the limitations of the experimental reference (same ODE model family) and about the ad hoc rejection filters. The comparison across three architectures and between ODE- and PDE-trained models is a useful empirical contribution. However, the central synthetic evaluation appears compromised by a train/test split that leaks near-duplicate inputs, and no code or data are provided for reproducibility. The clinical evaluation, while honestly caveated, does not provide independent validation of h and c predictions. The contribution is currently a promising but not fully validated framework.

major comments (3)
  1. [§3.1] The train/test split is performed on the augmented dataset, not on the underlying parameter samples. Each original input has two Gaussian-smoothed noisy copies, and all three are then randomly split 75/25. For a given parameter sample, the probability that at least one of its three copies lands in both training and test is 1 − (0.75³ + 0.25³) = 0.5625. Since the noise copies are small perturbations (σ=9, amplitude 0.25|I₁−I_N|), many test inputs are near-duplicates of training inputs. This measures denoising/interpolation rather than generalization to new parameter regimes, so the rRMSE values in Figs. 2 and 4 are likely optimistic. This is load-bearing for the central claim of a fast surrogate mapping. The fix is to split by parameter sample before augmentation, keeping all copies of a sample together in the same split.
  2. [§4.2] The experimental benchmark uses the fits from Driscoll et al. [23] as reference values. Those fits are obtained by solving an inverse problem for the same ODE model (Eq. 7) that generated the ODE training data, and the authors of [23] overlap with the current authors. The paper acknowledges this and calls PDE comparisons 'discrepancies,' but the abstract's claim that the method 'could be used in real time to predict characteristics of tear film flows from FL imaging' is not supported by any independent measurement of thickness or osmolarity. A concrete test would be to compare predictions on clinical traces against a validation method that does not use the same ODE family, such as a PDE-based fit or experimental thickness measurements, or at least to report the distribution of ODE fit residuals on the clinical traces.
  3. [§3.1] The rejection filters (h>1.1, h<0.2, h(1)>1.5·min h(t), and increasing I excluded) shape the training distribution, but the manuscript reports no counts of how many Halton samples were rejected, nor any diagnostic on whether the 467 clinical intensity traces satisfy these filters. If real TBU signals lie outside the surviving parameter set Ω, the learned map will be accurate on synthetic tests but systematically wrong on patients. This is a load-bearing threat to the clinical applicability claim. The authors should report rejection rates and check the clinical traces against the filter conditions (e.g., whether any clinical traces have increasing I or violate the h bounds in the reference fits).
minor comments (5)
  1. [Table 4] The column labels for ODE and PDE appear to be swapped. Per §2.2, the ODE parameters are h0, f0, ts, J_e, b1, b2, while the PDE parameters are h0, f0, ts, v', R_I, (Δσ)_0. The table currently shows R_I and (Δσ)_0 under ODE and b1, b2 under PDE, which contradicts the text. Please correct the table.
  2. [§3.1] The noise augmentation adds two perturbed copies per input with clean copies of the corresponding outputs. This setup is reasonable for denoising, but the manuscript should state explicitly that the output is always the clean solution, and clarify whether the test set includes noisy inputs with clean targets. This is relevant to interpretation of rRMSE.
  3. [§4.1] The statement 'Thickness h was predicted to at least one digit of accuracy in nearly all cases' should be revised after the train/test split is corrected; the current quantitative claim is based on the leaked split.
  4. [Figure 8] The caption 'By rows: osmolarity error to thickness error; By columns: PDE Dense-PCAX model to ODE Dense-PCAX model' is ambiguous. Specify which panel corresponds to which model and which error quantity is shown, including the color scale.
  5. [General] No code, data, or model weights are provided. For a data-driven paper of this type, providing the synthetic dataset generation scripts and trained model definition is important for reproducibility, especially because the rejection thresholds and noise augmentation are central to the results.

Circularity Check

0 steps flagged

No significant circularity: the operator learning pipeline is a standard supervised surrogate. Minor self-referential benchmarking against [23] is openly acknowledged; the train/test leakage is a statistical validity issue, not a derivation-level circularity.

full rationale

The paper's central claim is that a neural operator can approximate the inverse map I(t) -> (h,c) from simulations. This is supervised function approximation: training inputs are simulator-generated FL intensities and labels are simulator outputs; no equation of motion is derived from the learner and no parameter is fit to the test labels. The ODE model in (6)-(7) is inherited from [23] but restated explicitly, so there is no ansatz smuggled through citation; the exponential shear term is a stated modeling assumption. The only self-referential element is the experimental benchmark: the 467 clinical traces are labeled by ODE fits from [23] (author overlap), and the ODE-trained learners are trained on the same ODE model. The paper flags this itself: 'while we continue to report errors ... with respect to [23] as reference values, they might better be described as discrepancies for the PDE predictions' and notes the [23] fits are selected good fits (selection bias). This is an acknowledged limitation, not a load-bearing circularity. The skeptic's leakage concern is real but belongs to experimental validity, not circularity: §3.1 states two Gaussian-perturbed copies of every I vector were added before a random 75/25 split, so a typical parameter sample has copies spanning both splits; test items are then near-duplicates of training items, making synthetic rRMSE partly a denoising/interpolation score. The correct fix is to split by parameter sample. That flaw may inflate reported generalization, but it does not make the model's target definitionally equal to its input, so the circularity score remains low. Overall, the derivation chain is self-contained: forward models, sampling, and losses are all explicit, and the central result is an empirical surrogate, not a theorem derived from its conclusion.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central contribution is a learned surrogate, so the ledger is dominated by the simulation/domain assumptions that define the training data and by hand-set ML hyperparameters. No new physical entities are introduced.

free parameters (5)
  • Smoothness regularization α = 1.6
    Chosen by hand in §3.2 to suppress high-frequency oscillations; affects the loss landscape and thus the learned operator.
  • PCA component counts = 4 input; output 4 (ODE), 13 (PDE c), 8 (PDE h)
    Selected in §4 for reconstruction MSE ≈1.4e-4; dimensionality reduction is part of the learned mapping.
  • Fourier frequency scales = {0.5, 1, 2}
    Chosen in §3.2 to capture varying resolution; ad hoc.
  • Noise augmentation parameters = Gaussian smoothing σ=9, std 0.25|I1-IN|
    Set in §3.1 to mimic experimental noise; affects training distribution.
  • Rejection thresholds = h>1.1, h<0.2, h(1)>1.5 min h(t), increasing I
    Ad hoc filters in §3.1 that define the training distribution and therefore what the operator learns.
axioms (5)
  • domain assumption The 1D PDE system (1a)-(1e) from [19] accurately describes tear film breakup dynamics at the glob center.
    Used to generate the PDE training set; the paper does not validate this model against clinical data, only against prior modeling literature.
  • domain assumption The ODE model (7a)-(7d) from [23] is a valid simplification of tear film dynamics.
    Used to generate the ODE training set and as the reference model for experimental comparisons.
  • domain assumption Fluorescein intensity is given by the Webber-Jones relation I=I0(1-exp(-ϕ f h))/(1+f^2) (Eq. 5).
    Underpins the mapping from simulated h,f to synthetic intensity observations; errors here propagate to the learned operator.
  • standard math Neural networks of the described architectures can represent the I→(h,c) operator to the reported accuracy.
    Assumes universal approximation and that finite training data is representative; no formal guarantee is given for this specific operator.
  • ad hoc to paper The Halton-sampled parameter ranges (Table 4) and rejection filters produce a training distribution representative of clinical TBU intensity time series.
    The ranges and filters are chosen by the authors; if real clinical data falls outside these, predictions will be unreliable.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Operator learning for models of tear film breakup." pith.science (2026). https://pith.science/paper/BK27E3CI

@misc{pith2026260108001,
  author       = {Pith},
  title        = {Pith review of: Operator learning for models of tear film breakup},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BK27E3CI}},
  note         = {Machine review of arXiv:2601.08001}
}
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read the original abstract

Tear film (TF) breakup is a key driver of understanding dry eye disease, yet estimating TF thickness and osmolarity from fluorescence (FL) imaging typically requires solving computationally expensive inverse problems. We propose an operator learning framework that replaces traditional inverse solvers with neural operators trained on simulated TF dynamics. This approach offers a scalable path toward rapid, data-driven analysis of tear film dynamics.

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Doane, M.G.: Interaction of eyelids and tears in corneal wetting and the dynamics of the normal human eyeblink. Am. J. Ophthalmol.89, 507–516 (1980)

    1980

  2. [2]

    Willcox, M.D.P., Arg¨ ueso, P., Georgiev, G.A., Holopainen, J.M., Laurie, G.W., Millar, T.J., Papas, E.B., Rolland, J.P., Schmidt, T.A., Stahl, U., Suarez, T., Subbaraman, L.N., U¸ cakhan, O.¨O., Jones, L.: TFOS DEWS II Tear Film Report. Ocul. Surf.15(3), 366–403 (2017)

    2017

  3. [3]

    Braun, R.J., King-Smith, P.E., Begley, C.G., Li, L., Gewecke, N.R.: Dynamics and function of the tear film in relation to the blink cycle. Prog. Ret. Eye Res. 45, 132–164 (2015) 15 2.5 2.0 1.5 1.0 0.5 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Density FFN c h 2.5 2.0 1.5 1.0 0.5 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Dense-PCA 2.5 2.0 1.5 1.0 0.5 0.0...

    2015

  4. [4]

    Holly, F.J.: Formation and rupture of the tear film. Exp. Eye Res.15(5), 515–525 (1973)

    1973

  5. [5]

    King-Smith, P.E., Fink, B.A., Hill, R.M., Koelling, K.W., Tiffany, J.M.: The thickness of the tear film. Curr. Eye Res.29(4-5), 357–368 (2004)

    2004

  6. [6]

    Mishima, S., Maurice, D.M.: The oily layer of the tear film and evaporation from the corneal surface. Exp. Eye Res.1, 39–45 (1961)

    1961

  7. [7]

    Gipson, I.K.: Distribution of mucins at the ocular surface. Exp. Eye Res.78(3), 379–388 (2004)

    2004

  8. [8]

    Dartt, D.A.: Neural regulation of lacrimal gland secretory processes: Relevance in dry eye diseases. Prog. Ret. Eye Res.28(3), 155–177 (2009)

    2009

  9. [9]

    Braun, R.J.: Dynamics of the Tear Film. Annu. Rev. Fluid Mech.44(1), 267–297 16 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Je 1 0 1 2 3 b1 b1 vs Je (c error PDE) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Je 1 0 1 2 3 b1 b1 vs Je (c error ODE) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Je 1 0 1 2 3 b1 b1 vs Je (h error PDE) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Je 1 0 1 2 3 b1 b1 vs Je (h error ODE) 0.2 0.4 0...

    2012

  10. [10]

    The ocular surface16(1), 4–30 (2018)

    King-Smith, P.E., Begley, C.G., Braun, R.J.: Mechanisms, imaging and structure of tear film breakup. The ocular surface16(1), 4–30 (2018)

    2018

  11. [11]

    Lemp, MA.: The definition and classification of dry eye disease: Report of the Def- inition and Classification Subcommittee of the International Dry Eye WorkShop (2007). Ocul. Surf.5(2), 75–92 (2007)

    2007

  12. [12]

    Nelson, J.D., Craig, J.P., Akpek, E.K., Azar, D.T., Belmonte, C., Bron, A.J., Clayton, J.A., Dogru, M., Dua, H.S., Foulks, G.N., Gomes, J.A.P., Hammitt, K.M., Holopainen, J., Jones, L., Joo, C.-K., Liu, Z., Nichols, J.J., Nichols, K.K., Novack, G.D., Sangwan, V., Stapleton, F., Tomlinson, A., Tsubota, K., Willcox, M.D.P., Wolffsohn, J.S., Sullivan, D.A.: ...

    2017

  13. [13]

    Stapleton, F., Alves, M., Bunya, V.Y., Jalbert, I., Lekhanont, K., Malet, F., Na, K.-S., Schaumberg, D., Uchino, M., Vehof, J., Viso, E., Vitale, S., Jones, L.: TFOS DEWS-II epidemiology report. Ocul. Surf.15(3), 334–365 (2017)

    2017

  14. [14]

    Lin, H., Yiu, S.C.: Dry eye disease: A review of diagnostic approaches and treatments. Saudi J. Ophthalmol.28(3), 173–181 (2014)

    2014

  15. [15]

    Baudouin, C., Aragona, P., Messmer, E.M., Tomlinson, A., Calonge, M., Boboridis, K.G., Akova, Y.A., Geerling, G., Labetoulle, M., Rolando, M.: Role of hyperosmolarity in the pathogenesis and management of dry eye disease: Proceedings of the OCEAN group meeting. Ocul. Surf.11, 246–258 (2013)

    2013

  16. [16]

    Acta Ophthalmol.48(1), 108–118 (1970) 18

    Norn, M.S.: Micropunctate Fluorescein Vital Staining of the Cornea. Acta Ophthalmol.48(1), 108–118 (1970) 18

    1970

  17. [17]

    Peng, C.-C., Cerretani, C., Braun, R.J., Radke, C.J.: Evaporation-driven insta- bility of the precorneal tear film. Adv. Coll. Interface Sci.206, 250–264 (2014)

    2014

  18. [18]

    Braun, R.J., Driscoll, T.A., Begley, C.G., King-Smith, P.E., Siddique, J.I.: On tear film breakup (TBU): Dynamics and imaging. Math. Med. Biol.: A Journal of the IMA35(2), 145–180 (2018)

    2018

  19. [19]

    Zhong, L., Braun, R.J., Begley, C.G., King-Smith, P.E.: Dynamics of Fluorescent Imaging for Rapid Tear Thinning. Bull. Math. Biol.81(1), 39–80 (2019)

    2019

  20. [20]

    Luke, R.A., Braun, R.J., Driscoll, T.A., Begley, C.G., Awisi-Gyau, D.: Parameter Estimation for Evaporation-Driven Tear Film Thinning. Bull. Math. Biol.82(6), 71 (2020)

    2020

  21. [21]

    Luke, R.A., Braun, R.J., Driscoll, T.A., Awisi-Gyau, D., Begley, C.G.: Parameter Estimation for Mixed-Mechanism Tear Film Thinning. Bull. Math. Biol.83(5), 56 (2021)

    2021

  22. [22]

    Luke, R.A., Braun, R.J., Driscoll, T.A., Sinopoli, D., Yawatkar, V., You, L., Phatak, A., Begley, C.G.: Fitting Simplified Models to Machine Learning- Identified Tear Film Breakup. Invest. Ophthalmol. Vis. Sci.62(8), 1315–1315 (2021)

    2021

  23. [23]

    Driscoll, T.A., Braun, R.J., Luke, R.A., Sinopoli, D., Phatak, A., Dorsch, J., Begley, C.G., Awisi-Gyau, D.: Fitting ODE models of tear film breakup. Model. Artif. Intell. Ophthalmol.5, 1–36 (2023)

    2023

  24. [24]

    Physica D: Nonlinear Phenomena342, 1–15 (2017)

    Ji, H., Witelski, T.P.: Finite-time thin film rupture driven by modified evaporative loss. Physica D: Nonlinear Phenomena342, 1–15 (2017)

    2017

  25. [25]

    Shi, X., Fuller, G.G., Shaqfeh, E.S.G.: Instability and symmetry breaking of surfactant films over an air bubble. J. Fluid Mech.953, 26 (2022)

    2022

  26. [26]

    Journal of Engineering Mathematics149(1), 5 (2024) https://doi.org/10.1007/s10665-024-10407-6

    Chen, Q., Driscoll, T.A., Braun, R.J.: Evaporation-driven tear film thinning and breakup in two space dimensions. Journal of Engineering Mathematics149(1), 5 (2024) https://doi.org/10.1007/s10665-024-10407-6

    doi:10.1007/s10665-024-10407-6 2024

  27. [27]

    in preparation

    Chen, Q., Driscoll, T.A.: Parameter Estimation for an Evaporation-Driven Tear Film Model in Two Space Dimensions. in preparation

  28. [28]

    Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods

    Chen, C., Wu, J.-L.: Operator Learning for Continuous Spatial-Temporal Model with A Hybrid Optimization Scheme. arXiv (2023). https://doi.org/10.48550/ arXiv.2311.11798

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2311.11798 2023

  29. [29]

    Elsevier (2021)

    Wang, S., Wang, H., Perdikaris, P.: On the Eigenvector Bias of Fourier Feature Networks: From Regression to Solving Multi-Scale PDEs with Physics-Informed 19 Neural Networks. Elsevier (2021). https://doi.org/10.1016/j.cma.2021.113938 . https://doi.org/10.1016/j.cma.2021.113938 Accessed 2025-06-05

    arXiv 2021

  30. [30]

    In: Advances in Neural Information Processing Systems, vol

    Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Sing- hal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains. In: Advances in Neural Information Processing Systems, vol. 33, pp. 7537–7547. Curran Associates, Inc., Red Hook, NY, USA (2020)

    2020

  31. [31]

    Journal of Machine Learning Research24(318), 1–67 (2023)

    Lanthaler, S.: Operator learning with pca-net: upper and lower complexity bounds. Journal of Machine Learning Research24(318), 1–67 (2023)

    2023

  32. [32]

    Internat

    Tiffany, J.M.: The viscosity of human tears. Internat. Ophthalmol.15(6), 371–376 (1991)

    1991

  33. [33]

    Current Eye Research19(1), 4–11 (1999)

    Nagyov´ a, B., Tiffany, J.M.: Components responsible for the surface tension of human tears. Current Eye Research19(1), 4–11 (1999)

    1999

  34. [34]

    Journal of Colloid and Interface Science240(1), 259–271 (2001) https://doi.org/ 10.1006/jcis.2001.7562

    Ajaev, V.S., Homsy, G.M.: Steady vapor bubbles in rectangular microchannels. Journal of Colloid and Interface Science240(1), 259–271 (2001) https://doi.org/ 10.1006/jcis.2001.7562

    arXiv 2001

  35. [35]

    Casalini, T., Salvalaglio, M., Perale, G., Masi, M., Cavallotti, C.: Diffusion and aggregation of sodium fluorescein in aqueous solutions. J. Phys. Chem.. B 115(44), 12896–12904 (2011)

    2011

  36. [36]

    Riquelme, R., Lira, I., P´ erez-L´ opez, C., Rayas, J.A., Rodr ´ ıguez-Vera, R.: Interfer- ometric measurement of a diffusion coefficient: Comparison of two methods and uncertainty analysis. J. Phys. D: Appl. Phys.40(9), 2769–2776 (2007)

    2007

  37. [37]

    Lemp, M.A., Bron, A.J., Baudouin, C., Ben ´ ıtez Del Castillo, J.M., Geffen, D., Tauber, J., Foulks, G.N., Pepose, J.S., Sullivan, B.D.: Tear osmolarity in the diagnosis and management of dry eye disease. Am.J. Ophthalmol.151(5), 792– 7981 (2011)

    2011

  38. [38]

    Determination of molar absorption coefficient

    Mota, M.C., Carvalho, P., Ramalho, J., Leite, E.: Spectrophotometric analy- sis of sodium fluorescein aqueous solutions. Determination of molar absorption coefficient. Internat. Ophthalmol.15(5), 321–326 (1991)

    1991

  39. [39]

    Webber, W.R., Jones, D.P.: Continuous fluorophotometric method of measuring tear turnover rate in humans and analysis of factors affecting accuracy. Med. Biol. Eng. Comput.24(4), 386–392 (1986)

    1986

  40. [40]

    Braun, R.J., Gewecke, N.R., Begley, C.G., King-Smith, P.E., Siddique, J.I.: A Model for Tear Film Thinning With Osmolarity and Fluorescein. Invest. Ophthalmol. Vis. Sci.55(2), 1133–1142 (2014) 20

    2014

  41. [41]

    Wu, Z., Begley, C.G., Port, N., Bradley, A., Braun, R., King-Smith, E.: The Effects of Increasing Ocular Surface Stimulation on Blinking and Tear Secretion. Invest. Ophthalmol. Vis. Sci.56(8), 4211–4220 (2015)

    2015

  42. [42]

    Software, Environments, Tools

    Trefethen, L.N.: Spectral Methods in Matlab. Software, Environments, Tools. SIAM, Philadelphia (2000)

    2000

  43. [43]

    Journal of Open Research Software5(1), 15 (2017)

    Rackauckas, C., Nie, Q.: Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia. Journal of Open Research Software5(1), 15 (2017)

    2017

  44. [44]

    Wang, S., Wang, H., Perdikaris, P.: On the eigenvector bias of fourier feature net- works: From regression to solving multi-scale pdes with physics-informed neural networks. Comput. Methods Appl. Mech. Eng.384, 113938 (2021)

    2021

  45. [45]

    Yadav, S.: Combining pca with neural networks: Improving model efficiency and interpretability. Int. J. Innov. Res. Eng. Multidiscip. Phys. Sci.7, (2019)

    2019

  46. [46]

    Journal of Scientific Computing92(3), 88 (2022) https: //doi.org/10.1007/s10915-022-01939-z

    Cuomo, S., Di Cola, V.S., Giampaolo, F., Rozza, G., Raissi, M., Piccialli, F.: Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next. Journal of Scientific Computing92(3), 88 (2022) https: //doi.org/10.1007/s10915-022-01939-z

    doi:10.1007/s10915-022-01939-z 2022

  47. [47]

    arXiv (2023)

    Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., Anandkumar, A.: Physics-Informed Neural Operator for Learning Partial Differential Equations. arXiv (2023). https://doi.org/10.48550/arXiv.2111.03794

    Pith/arXiv arXiv doi:10.48550/arxiv.2111.03794 2023

  48. [48]

    arXiv (2019)

    Chen, R.T.Q., Rubanova, Y., Bettencourt, J., Duvenaud, D.: Neural Ordinary Differential Equations. arXiv (2019). https://doi.org/10.48550/arXiv.1806.07366

    Pith/arXiv arXiv doi:10.48550/arxiv.1806.07366 2019

  49. [49]

    arXiv (2022)

    Kidger, P.: On Neural Differential Equations. arXiv (2022). https://doi.org/10. 48550/arXiv.2202.02435

    Pith/arXiv arXiv doi:10.48550/arxiv.2202.02435 2022

  50. [50]

    arXiv:2001.04385 [cs, math, q-bio, stat] (2020) arxiv:2001.04385 [cs, math, q-bio, stat] 21

    Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., Skin- ner, D., Ramadhan, A.: Universal Differential Equations for Scientific Machine Learning. arXiv:2001.04385 [cs, math, q-bio, stat] (2020) arxiv:2001.04385 [cs, math, q-bio, stat] 21

    Pith/arXiv arXiv 2001

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.