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REVIEW 3 major objections 3 minor 1 cited by

Rethinking Multimodality: Optimizing Multimodal Deep Learning for Biomedical Signal Classification

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that for ECG classification, fusing time-domain and time-frequency-domain models beats any single-domain baseline, while adding the frequency domain as a third modality produces no further gain.

desk verdict The abstract is a plausible ECG multimodal paper, but the body is an unrelated material-property paper, so there is no experiment to review. read the letter →

arxiv 2508.00963 v1 pith:SHLFHY3E submitted 2025-08-01 cs.LG cs.AI

Timothy Oladunni , Alex Wong This is my paper · ORCID
classification cs.LGcs.AI
keywords multimodaldeeplearningECGclassificationcomplementaryfeaturedomainstime-frequencyanalysisrepresentationalredundancyhybridneuralnetworksBayesianinferenceablationstudy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that adding more input domains to a multimodal deep learning model does not automatically improve ECG classification; what matters is whether the domains carry complementary information. The authors build five models—three single-domain encoders and two fusions—and compare them on ECG classification with bootstrapping and Bayesian inference. They find that the two-domain hybrid fusing raw time with time-frequency representations consistently outperforms the best single-domain baseline, while adding a third frequency-domain stream gives no further gain and sometimes a small loss. The intended upshot is a design principle: optimal multimodal performance comes from information-theoretic complementarity between fused domains, not from the number of modalities.

What carries the argument

The argument runs through five architectures: a 1D-CNN on raw time signals, a 2D-CNN on time-frequency representations, a 1D-CNN-Transformer (an attention-based sequence model) on frequency spectra, Hybrid 1 (1D-CNN + 2D-CNN), and Hybrid 2 (1D-CNN + 2D-CNN + Transformer). The load-bearing device is the comparison between Hybrid 1 and Hybrid 2, evaluated with bootstrapping and Bayesian inference so that the difference is assessed as a distribution rather than a point estimate. The named framework, 'Complementary Feature Domains in Multimodal ECG Deep Learning,' is the mathematical account the paper offers for why the time and time-frequency domains are synergistic while the frequency domain is redundant. The ablation study is what connects the performance gap to representational redundancy rather than to the extra parameters of Hybrid 2.

What would settle it

Train Hybrid 2 with a frequency encoder of strictly greater capacity, or with matched parameter count and optimization schedule, and check whether the gap with Hybrid 1 closes; if it does, the claimed redundancy is an artifact of the encoder rather than the domain. A complementary check is to compute the mutual information between the learned features of the time-frequency and frequency encoders: high mutual information with no accuracy gain would support redundancy, while low mutual information with no gain would undermine it.

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Extended reading notes

Core claim

The central claim is that complementarity, not modality count, determines the value of multimodal fusion for biomedical signal classification. On ECG data, the time-domain 1D-CNN and the time-frequency 2D-CNN are complementary: their fusion (Hybrid 1) beats the 2D-CNN baseline on every reported metric, with p-values below 0.05 and Bayesian probabilities above 0.90. The frequency-domain 1D-CNN-Transformer does not add complementary information; when it is appended to form Hybrid 2, performance does not improve and can slightly decline. The paper attributes this to representational redundancy between the frequency and time-frequency domains, and reports a targeted ablation study supporting that explanation. It generalizes the finding into a proposed framework, 'Complementary Feature Domains in Multimodal ECG Deep Learning,' intended to quantify which domain combinations are ideal.

Load-bearing premise

The comparison assumes the three single-domain encoders are equally well tuned and of comparable learning capacity, so the frequency encoder's lack of benefit reflects redundancy in the data rather than a weaker model.

Editorial extensions

If this is right

  • Multimodal ECG models should be built by pairing domains with complementary information, not by stacking every available representation.
  • A three-domain hybrid with redundant domains can underperform a two-domain hybrid despite seeing strictly more input data.
  • Bootstrapping and Bayesian inference provide a usable protocol for deciding whether an added modality earns its place in a biomedical classifier.
  • The 'Complementary Feature Domains' principle gives a quantitative way to rank candidate domain combinations before training.
  • Future multimodal fusion studies should report whether each added domain improves accuracy beyond the best single domain, rather than only comparing fused models against weaker baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the paper's reasoning would be to hold one encoder fixed and vary the second domain, e.g., replace the frequency spectrum with heart-rate variability features, to see whether complementarity rather than the specific domain drives the gain.
  • The complementarity criterion could be measured directly: compute mutual information between latent representations of candidate encoders before training and test whether that score predicts which fusion performs best.
  • The same 'redundant third domain' pattern likely appears in other physiological signals where time, spectral, and time-frequency views are routinely fused, such as EEG or PPG.
  • If the framework is right, the practical cost of multimodal systems can be cut by pruning redundant streams before training, without sacrificing accuracy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The manuscript, as submitted, consists of an abstract claiming a multimodal deep learning study for ECG classification and a full text that is an unrelated paper on embedding-oriented Physically-Guided Neural Networks with Internal Variables for material behavior discovery. The abstract describes three unimodal encoders (1D-CNN, 2D-CNN, 1D-CNN-Transformer), two hybrid fusions, bootstrap and Bayesian analyses, and a proposed "Complementary Feature Domains" framework. The full text contains no ECG dataset, no 1D-CNN/2D-CNN/Transformer encoders, no Hybrid 1 or Hybrid 2 architectures, and no bootstrap or Bayesian inference results. The claimed framework is not defined anywhere in the submitted text. Consequently, the central claim that Hybrid 1 outperforms the 2D-CNN baseline due to domain complementarity while Hybrid 2 adds redundancy cannot be checked against any evidence in the manuscript.

Significance. If the ECG experiments described in the abstract were fully reported, the empirical finding that fusing time and time-frequency domains helps while adding a redundant frequency domain does not help could be a useful, if modest, contribution to multimodal biomedical signal classification. The proposed theoretical framework, however, has no formal statement in the submitted text, so its significance cannot currently be assessed. The full text is a separate, more developed SciML paper with its own open-source repository, but that content is not the paper claimed by the abstract and does not bear on the ECG claim. As submitted, the manuscript is not a coherent article and the central contribution is unverified.

major comments (3)
  1. [Abstract vs. full text] The full text (Sections 1–6) is titled "Enhancing material behavior discovery using embedding-oriented Physically-Guided Neural Networks with Internal Variables" and addresses a nonlinear diffusion problem with spectral, POD, and autoencoder decoders. It contains no ECG classification experiment, no 1D-CNN, 2D-CNN, or Transformer encoders, no Hybrid 1 or Hybrid 2 architectures, and no bootstrap or Bayesian results. The p-values and Bayesian probabilities cited in the abstract therefore have no supporting experiment in the manuscript. This is a load-bearing absence: the central claim that Hybrid 1 outperforms the 2D-CNN baseline is unsupported by the submitted text and cannot be fixed by local edits.
  2. [Abstract (framework definition)] The abstract asserts a "mathematically quantifiable framework" named "Complementary Feature Domains in Multimodal ECG Deep Learning" and invokes "intrinsic information-theoretic complementarity," but the submitted text provides no definition, equations, or formal criterion for complementarity. As written, the framework risks being circular: if complementarity is measured by the same performance differences it is invoked to explain, then the conclusion is definitional rather than explanatory. The authors need to state an independent measure of complementarity—for example, an information-theoretic quantity computed from the learned representations—and show how Hybrid 1's advantage follows from that measure.
  3. [Abstract (statistical evidence)] Even taken solely on its own terms, the abstract reports p-values and Bayesian probabilities without effect sizes, confidence intervals, dataset description, train/validation split details, model capacity matching, or multiple-testing corrections. Because the full text provides none of these details, the reported statistical evidence cannot be independently checked. A complete experimental section with architecture specifications, hyperparameters, capacity-matched baselines, and full result tables is required before the claimed findings can be evaluated.
minor comments (3)
  1. [Title and authorship consistency] The abstract and full text have different titles and clearly describe different research areas; the manuscript must be made internally consistent or resubmitted with the correct body.
  2. [Terminology] Phrases such as "paradigm-shifting" and "rigorously evaluated" in the abstract are not supported by the submitted content and should be replaced with specific, measurable claims.
  3. [Reproducibility] The full text points to a GitHub repository for the PGNNIV study, but no repository, dataset, or code is provided for the ECG experiments claimed in the abstract; such artifacts are necessary for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity can be established because the submitted body is an unrelated material-behavior paper, leaving the ECG multimodal claims without any derivation chain.

full rationale

The submitted full text is a different manuscript, titled 'Enhancing material behavior discovery using embedding-oriented physically-guided neural networks with internal variables.' It contains no ECG dataset, no 1D-CNN/2D-CNN/Transformer encoders, no Hybrid 1 or Hybrid 2 fusion architectures, and no bootstrap or Bayesian inference results. The abstract's central empirical claims (p-values < 0.05, Bayesian probabilities > 0.90) therefore have no experimental or derivational support in the body, which is a serious missing-support defect rather than a demonstrated circularity. Under the hard rule that circularity requires quoting a specific reduction (e.g., Eq. X = Eq. Y by construction, or a fitted parameter renamed as a prediction), I cannot exhibit such a reduction because the relevant framework and experiments are absent. The only candidate circularity would be defining 'complementarity' by the performance differences it is then used to explain, but the abstract gives no such definition and no equations, so that equivalence cannot be shown from the text. Self-citations in the body are to prior PGNNIV work, but the new embedding and transfer-learning contributions are evaluated against those prior models as baselines, not derived from them by definition. The honest finding is therefore no significant circularity, score 0, while emphasizing that the manuscript does not support its stated central claim.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only review; no free parameters or entities can be identified. The axioms listed are the implicit premises on which the abstract's conclusions depend.

assumptions (3)
  • domain assumption Time, time-frequency, and frequency representations of ECG signals capture distinct information; these domains are appropriate input views for the fusion models.
    The entire comparison rests on the choice of these three domains as representative unimodal views. This is implicit in the model design but not justified in the abstract.
  • standard math Bootstrapping and Bayesian inference provide valid error metrics for comparing classifier performance on the ECG test data.
    The abstract reports p-values and Bayesian probabilities, assuming these statistical tools are correctly applied. No details on the statistical procedure are given.
  • ad hoc to paper The proposed 'Complementary Feature Domains' framework provides a mathematically valid quantification of complementarity.
    This is the paper's central new concept, asserted in the abstract but not defined. Its validity is assumed by the conclusion that complementarity, not modality count, drives performance.

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Cite this review

Pith. "Pith review of Rethinking Multimodality: Optimizing Multimodal Deep Learning for Biomedical Signal Classification." pith.science (2026). https://pith.science/paper/SHLFHY3E

@misc{pith2026250800963,
  author       = {Pith},
  title        = {Pith review of: Rethinking Multimodality: Optimizing Multimodal Deep Learning for Biomedical Signal Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHLFHY3E}},
  note         = {Machine review of arXiv:2508.00963}
}
read the original abstract

This study proposes a novel perspective on multimodal deep learning for biomedical signal classification, systematically analyzing how complementary feature domains impact model performance. While fusing multiple domains often presumes enhanced accuracy, this work demonstrates that adding modalities can yield diminishing returns, as not all fusions are inherently advantageous. To validate this, five deep learning models were designed, developed, and rigorously evaluated: three unimodal (1D-CNN for time, 2D-CNN for time-frequency, and 1D-CNN-Transformer for frequency) and two multimodal (Hybrid 1, which fuses 1D-CNN and 2D-CNN; Hybrid 2, which combines 1D-CNN, 2D-CNN, and a Transformer). For ECG classification, bootstrapping and Bayesian inference revealed that Hybrid 1 consistently outperformed the 2D-CNN baseline across all metrics (p-values < 0.05, Bayesian probabilities > 0.90), confirming the synergistic complementarity of the time and time-frequency domains. Conversely, Hybrid 2's inclusion of the frequency domain offered no further improvement and sometimes a marginal decline, indicating representational redundancy; a phenomenon further substantiated by a targeted ablation study. This research redefines a fundamental principle of multimodal design in biomedical signal analysis. We demonstrate that optimal domain fusion isn't about the number of modalities, but the quality of their inherent complementarity. This paradigm-shifting concept moves beyond purely heuristic feature selection. Our novel theoretical contribution, "Complementary Feature Domains in Multimodal ECG Deep Learning," presents a mathematically quantifiable framework for identifying ideal domain combinations, demonstrating that optimal multimodal performance arises from the intrinsic information-theoretic complementarity among fused domains.

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Reference graph

Works this paper leans on

51 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    Virtual, digital and hybrid twins: a new paradigm in data-based engineering and engineered data.Archives of computational methods in engineering, 27:105–134, 2020

    Francisco Chinesta, Elias Cueto, Emmanuelle Abisset-Chavanne, Jean Louis Duval, and Fouad El Khaldi. Virtual, digital and hybrid twins: a new paradigm in data-based engineering and engineered data.Archives of computational methods in engineering, 27:105–134, 2020

    work page 2020

  2. [2]

    Empowering engineering with data, machine learning and artificial in- telligence: a short introductive review

    Francisco Chinesta and Elias Cueto. Empowering engineering with data, machine learning and artificial in- telligence: a short introductive review. Advanced Modeling and Simulation in Engineering Sciences , 9(1):21, 2022

    work page 2022

  3. [3]

    Computational sensing, understanding, and reasoning: an artificial intelligence approach to physics-informed world modeling

    Beatriz Moya, Alberto Badías, David González, Francisco Chinesta, and Elias Cueto. Computational sensing, understanding, and reasoning: an artificial intelligence approach to physics-informed world modeling. Archives of Computational Methods in Engineering, 31(4):1897–1914, 2024

    work page 1914

  4. [4]

    Deep learning in computational mechanics: a review.Computational Mechanics, 74(2):281–331, 2024

    Leon Herrmann and Stefan Kollmannsberger. Deep learning in computational mechanics: a review.Computational Mechanics, 74(2):281–331, 2024

    2024

  5. [5]

    Physics- informed machine learning

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics- informed machine learning. Nature Reviews Physics, 3(6):422–440, 2021

    2021

  6. [6]

    Physical laws meet machine intelligence: current developments and future directions

    Temoor Muther, Amirmasoud Kalantari Dahaghi, Fahad Iqbal Syed, and Vuong Van Pham. Physical laws meet machine intelligence: current developments and future directions. Artificial Intelligence Review, 56(7):6947–7013, 2023

    work page 2023

  7. [7]

    Ai meets physics: a comprehensive survey

    Licheng Jiao, Xue Song, Chao You, Xu Liu, Lingling Li, Puhua Chen, Xu Tang, Zhixi Feng, Fang Liu, Yuwei Guo, et al. Ai meets physics: a comprehensive survey. Artificial Intelligence Review, 57(9):256, 2024

    work page 2024

  8. [8]

    Knowledge integration into deep learning in dynamical systems: an overview and taxonomy

    Sung Wook Kim, Iljeok Kim, Jonghwan Lee, and Seungchul Lee. Knowledge integration into deep learning in dynamical systems: an overview and taxonomy. Journal of Mechanical Science and Technology, 35:1331–1342, 2021

    work page 2021

Show all 51 references
  1. [9]

    Mathematical models in science and engineering

    Alfio Quarteroni. Mathematical models in science and engineering. Notices of the AMS, 56(1):10–19, 2009

    2009

  2. [10]

    Introducing machine learning: science and technology

    O Anatole von Lilienfeld. Introducing machine learning: science and technology. Machine Learning: Science and Technology, 1(1):010201, 2020

    2020

  3. [11]

    Physics-informed machine learning and its structural integrity applications: state of the art

    Shun-Peng Zhu, Lanyi Wang, Changqi Luo, José AFO Correia, Abílio MP De Jesus, Filippo Berto, and Qingyuan Wang. Physics-informed machine learning and its structural integrity applications: state of the art. Philosophical Transactions of the Royal Society A, 381(2260):20220406, 2023

    2023

  4. [12]

    Physics-informed machine learning for reliability and systems safety applications: State of the art and challenges

    Yanwen Xu, Sara Kohtz, Jessica Boakye, Paolo Gardoni, and Pingfeng Wang. Physics-informed machine learning for reliability and systems safety applications: State of the art and challenges. Reliability Engineering & System Safety, 230:108900, 2023

    2023

  5. [13]

    Physics-informed machine learning for structural health monitoring

    Elizabeth J Cross, Samuel J Gibson, Matthew R Jones, Daniel J Pitchforth, Sikai Zhang, and Timothy J Rogers. Physics-informed machine learning for structural health monitoring. Structural health monitoring based on data science techniques, pages 347–367, 2022

    2022

  6. [14]

    Physics-informed machine learning: A comprehensive review on applications in anomaly detection and condition monitoring

    Yuandi Wu, Brett Sicard, and Stephen Andrew Gadsden. Physics-informed machine learning: A comprehensive review on applications in anomaly detection and condition monitoring. Expert Systems with Applications, page 124678, 2024

    2024

  7. [15]

    Physics-informed machine learning: case studies for weather and climate modelling

    Karthik Kashinath, M Mustafa, Adrian Albert, JL Wu, C Jiang, Soheil Esmaeilzadeh, Kamyar Azizzadenesheli, R Wang, Ashesh Chattopadhyay, A Singh, et al. Physics-informed machine learning: case studies for weather and climate modelling. Philosophical Transactions of the Royal So...

    2021

  8. [16]

    Machine learning for the physics of climate

    Annalisa Bracco, Julien Brajard, Henk A Dijkstra, Pedram Hassanzadeh, Christian Lessig, and Claire Monteleoni. Machine learning for the physics of climate. Nature Reviews Physics, 7(1):6–20, 2025

    2025

  9. [17]

    Recent advances and applications of machine learning in experimental solid mechanics: A review

    Hanxun Jin, Enrui Zhang, and Horacio D Espinosa. Recent advances and applications of machine learning in experimental solid mechanics: A review. Applied Mechanics Reviews, 75(6):061001, 2023

    2023

  10. [18]

    A review of physics-informed machine learning in fluid mechanics

    Pushan Sharma, Wai Tong Chung, Bassem Akoush, and Matthias Ihme. A review of physics-informed machine learning in fluid mechanics. Energies, 16(5):2343, 2023

    2023

  11. [19]

    Physics-guided, physics-informed, and physics-encoded neural networks and operators in scientific computing: Fluid and solid mechanics

    Salah A Faroughi, Nikhil M Pawar, Célio Fernandes, Maziar Raissi, Subasish Das, Nima K Kalantari, and Seyed Kourosh Mahjour. Physics-guided, physics-informed, and physics-encoded neural networks and operators in scientific computing: Fluid and solid mechanics. Journal of Compu...

    2024

  12. [20]

    Scientific machine learning through physics–informed neural networks: Where we are and what’s next

    Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing, 92(3):88, 2022

    2022

  13. [21]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707, 2019

    2019

  14. [22]

    Structure-preserving neural networks

    Quercus Hernández, Alberto Badías, David González, Francisco Chinesta, and Elías Cueto. Structure-preserving neural networks. Journal of Computational Physics, 426:109950, 2021

    2021

  15. [23]

    Thermodynamics-informed graph neural networks

    Quercus Hernández, Alberto Badías, Francisco Chinesta, and Elías Cueto. Thermodynamics-informed graph neural networks. IEEE Transactions on Artificial Intelligence, 5(3):967–976, 2022

    2022

  16. [24]

    Neural operator: Learning maps between function spaces with applications to pdes

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research, 24(89):1–97, 2023

    2023

  17. [25]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020

    2010 arXiv

  18. [26]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218–229, 2021

    2021

  19. [27]

    Nn-euclid: Deep-learning hyperelasticity without stress data

    Prakash Thakolkaran, Akshay Joshi, Yiwen Zheng, Moritz Flaschel, Laura De Lorenzis, and Siddhant Kumar. Nn-euclid: Deep-learning hyperelasticity without stress data. Journal of the Mechanics and Physics of Solids, 169:105076, 2022

    2022

  20. [28]

    Discovering plasticity models without stress data

    Moritz Flaschel, Siddhant Kumar, and Laura De Lorenzis. Discovering plasticity models without stress data. npj Computational Materials, 8(1):91, 2022

    2022

  21. [29]

    Automated discovery of generalized standard material models with euclid

    Moritz Flaschel, Siddhant Kumar, and Laura De Lorenzis. Automated discovery of generalized standard material models with euclid. Computer Methods in Applied Mechanics and Engineering, 405:115867, 2023

    2023

  22. [30]

    Prediction and identification of physical systems by means of physically-guided neural networks with meaningful internal layers

    Jacobo Ayensa-Jiménez, Mohamed H Doweidar, Jose A Sanz-Herrera, and Manuel Doblaré. Prediction and identification of physical systems by means of physically-guided neural networks with meaningful internal layers. Computer Methods in Applied Mechanics and Engineering, 381:113816, 2021

    2021

  23. [31]

    Predicting and explaining nonlinear material response using deep physically guided neural networks with internal variables

    Jacobo Ayensa-Jiménez, Javier Orera-Echeverría, and Manuel Doblare. Predicting and explaining nonlinear material response using deep physically guided neural networks with internal variables. Mathematics and Mechanics of Solids, 30(2):573–598, 2025

    2025

  24. [32]

    On the application of physically-guided neural networks with internal variables to continuum problems

    Rubén Muñoz-Sierra, Jacobo Ayensa-Jiménez, and Manuel Doblaré. On the application of physically-guided neural networks with internal variables to continuum problems. Mechanics of Materials, 205:105317, 2025

    2025

  25. [33]

    Understanding glioblastoma invasion using physically-guided neural networks with internal variables

    Jacobo Ayensa-Jiménez, Mohamed H Doweidar, Jose A Sanz-Herrera, and Manuel Doblare. Understanding glioblastoma invasion using physically-guided neural networks with internal variables. PLoS Computational Biology, 18(4):e1010019, 2022

    2022

  26. [34]

    Recent advances on the use of separated representations

    David González, Amine Ammar, Francisco Chinesta, and Elías Cueto. Recent advances on the use of separated representations. International Journal for Numerical Methods in Engineering, 81(5):637–659, 2010

    2010

  27. [35]

    Why and when can deep-but not shallow-networks avoid the curse of dimensionality: a review

    Tomaso Poggio, Hrushikesh Mhaskar, Lorenzo Rosasco, Brando Miranda, and Qianli Liao. Why and when can deep-but not shallow-networks avoid the curse of dimensionality: a review. International Journal of Automation and Computing, 14(5):503–519, 2017

    2017

  28. [36]

    Conceptual and empirical comparison of dimensionality reduction algorithms (pca, kpca, lda, mds, svd, lle, isomap, le, ica, t-sne)

    Farzana Anowar, Samira Sadaoui, and Bassant Selim. Conceptual and empirical comparison of dimensionality reduction algorithms (pca, kpca, lda, mds, svd, lle, isomap, le, ica, t-sne). Computer Science Review, 40:100378, 2021

    2021

  29. [37]

    Feature dimensionality reduction: a review

    Weikuan Jia, Meili Sun, Jian Lian, and Sujuan Hou. Feature dimensionality reduction: a review. Complex & Intelligent Systems, 8(3):2663–2693, 2022

    2022

  30. [38]

    John P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Books on Mathematics. Dover Publications, Mineola, NY , second edition, 2001

    2001

  31. [39]

    The homogeneous chaos

    Norbert Wiener. The homogeneous chaos. American Journal of Mathematics, 60(4):897–936, 1938

    1938

  32. [40]

    An introduction to the proper orthogonal decomposition

    Anindya Chatterjee. An introduction to the proper orthogonal decomposition. Current science, pages 808–817, 2000. 27 Enhancing material behavior discovery using embedding-oriented Physically-Guided Neural Networks with Internal Variables

    2000

  33. [41]

    On interpretability and proper latent decomposition of autoencoders

    Luca Magri and Anh Khoa Doan. On interpretability and proper latent decomposition of autoencoders. arXiv preprint arXiv:2211.08345, 2022

    2022 arXiv

  34. [42]

    An introduction to harmonic analysis

    Yitzhak Katznelson. An introduction to harmonic analysis. Cambridge University Press, 2004

    2004

  35. [43]

    W. Rudin. Functional Analysis. Higher Mathematics Series. McGraw-Hill, 1973

    1973

  36. [44]

    Cooley and John W

    James W. Cooley and John W. Tukey. An algorithm for the machine calculation of complex fourier series. Mathematics of Computation, 19(90):297–301, 1965

    1965

  37. [45]

    Code verification by the method of manufactured solutions

    Patrick J Roache. Code verification by the method of manufactured solutions. J. Fluids Eng., 124(1):4–10, 2002

    2002

  38. [46]

    PyTorch Documentation

    PyTorch Core Team. PyTorch Documentation. PyTorch Foundation, 2025

    2025

  39. [47]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. CoRR, abs/1412.6980, 2014

    2014 arXiv

  40. [48]

    Robust nonparametric statistical methods

    Thomas P Hettmansperger and Joseph W McKean. Robust nonparametric statistical methods. CRC press, 2010

    2010

  41. [49]

    Self-adaptive loss balanced physics-informed neural networks

    Zixue Xiang, Wei Peng, Xu Liu, and Wen Yao. Self-adaptive loss balanced physics-informed neural networks. Neurocomputing, 496:11–34, 2022

    2022

  42. [50]

    So (3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials

    Yousef Heider, Kun Wang, and WaiChing Sun. So (3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials. Computer Methods in Applied Mechanics and Engineering, 363:112875, 2020

    2020

  43. [51]

    Convex neural networks

    Yoshua Bengio, Nicolas Roux, Pascal Vincent, Olivier Delalleau, and Patrice Marcotte. Convex neural networks. Advances in neural information processing systems, 18, 2005. 28

    2005

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