Pith. sign in

REVIEW 3 major objections 5 minor 65 references

Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A two-stage network combining a Koopman-enhanced graph convolutional encoder with a Transformer decoder is claimed to keep cardiac excitation forecasts accurate on a 3D mesh for up to 300 time steps.

desk verdict A clean new combination of known pieces for cardiac mesh forecasting, but the paper's own ablation table contradicts its central superiority claim. read the letter →

arxiv 2507.03855 v1 pith:FFQSEGAN submitted 2025-07-05 cs.LG stat.ML

Zekai Wang , Bing Yao This is my paper
classification cs.LGstat.ML
keywords Transformerself-attentionmechanismgraphconvolutionalnetworkKoopmanoperatorspatiotemporaldynamicsforecastingcardiacelectrodynamicsablationstudy
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

TK-GCN is a proposed two-stage architecture for forecasting spatiotemporal dynamics on irregular 3D geometries, tested on simulated cardiac electrical-wave propagation. The paper argues that a graph-convolutional autoencoder should be trained jointly with a learnable Koopman matrix so that the high-dimensional voltage state maps into a latent space where time evolution is approximately linear; a frozen-encoder Transformer decoder then forecasts the latent trajectory and maps it back to the mesh. The intended claim is that this combination yields the most accurate forecasts across all tested horizons and protocols, and that ablations show each of the three components—Koopman operator, GCN, and Transformer—is needed. If true, this would give a reusable recipe for long-horizon forecasting of nonlinear, geometry-bound dynamics.

What carries the argument

The load-bearing object is the learned Koopman transition matrix $K\in\mathbb{R}^{d_z\times d_z}$, trained with a dynamics loss $L_{\rm dyn}$ that forces $\psi_e(x(t+\Delta t))\approx K^{\Delta t}\psi_e(x(t))$; this is what linearizes the latent evolution. The second component is the masked multi-head self-attention Transformer decoder that forecasts sequences of these latent vectors and is trained with a forecasting loss in latent space. The GCN provides graph pooling and unpooling with continuous B-spline kernels so the geometry of the mesh is respected throughout.

What would settle it

Run each model in Table 1 (TK-GCN, Transformer+GCN, Transformer Only, Transformer+CNN-AE, and LSTM+K-GCN) under the three protocols with at least ten random seeds and compare the full MSE distributions on the [0,100), [100,200), and [200,300) intervals. The claim is falsified if TK-GCN is not the best in every cell, or if any difference falls within seed noise—as the current single-run numbers already show Transformer Only winning Protocol I [0,100) and Transformer+GCN winning Protocol II [0,100) and [100,200).

Watch

Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is that nonlinear spatiotemporal dynamics on an irregular mesh can be split into a spatial encoding problem and a near-linear temporal evolution problem. A Koopman-enhanced graph convolutional autoencoder makes the latent states advance approximately linearly under a learned matrix, and a Transformer decoder then predicts those latent states over hundreds of steps; decoding returns voltage maps. The paper reports that this two-stage pipeline predicts simulated cardiac activation more accurately than LSTM, Dynamic Mode Decomposition, a pure Koopman operator, and vector autoregression across all three stimulation protocols, with the full architecture beating every ablation variant.

Load-bearing premise

The paper's conclusion rests on the assumption that the accuracy differences shown in its ablation table are real and repeatable, even though each number comes from a single run with no reported variance, seed count, or significance test.

Editorial extensions

If this is right

  • If the central claim holds, long-horizon forecasting on irregular 3D domains becomes practical by first linearizing latent dynamics, which reduces error accumulation in multi-step rollouts.
  • The ablation results imply the Koopman term is not a minor addition: the gap between TK-GCN and the same model without Koopman widens in mid- and long-term intervals, especially under Protocol I and Protocol III.
  • The framework is defined on any graph or mesh, so the conclusion draws immediate applications to traffic networks, environmental monitoring, and biological processes where regular grids are unnatural.
  • A fixed pretrained encoder can be reused with different temporal forecasters, since the Transformer only sees latent vectors and does not need to be retrained with the geometry encoder.

Reading between the lines

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

  • The paper's own Table 1 shows simpler variants beating TK-GCN in some short-horizon cells (Transformer Only under Protocol I in [0,100), and Transformer+GCN under Protocol II in [0,100) and [100,200)), which suggests the architecture's advantage may be specifically long-horizon stability rather than uniform superiority.
  • Because the data come from the same Aliev–Panfilov model that defines the smooth and chaotic regimes, the natural next test is whether the same pipeline survives noisy or real cardiac recordings; the paper does not attempt this.
  • A multi-seed evaluation with confidence intervals would be the decisive check; until then the rank ordering in individual cells should be read as directional evidence rather than proof of consistent dominance.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes TK-GCN, a two-stage framework for spatiotemporal forecasting on irregular domains. Stage 1 is a Koopman-enhanced graph convolutional network (K-GCN) autoencoder that learns a low-dimensional latent space in which the dynamics are approximately linear under a learned Koopman operator K. Stage 2 is a Transformer decoder that forecasts future latent vectors, which are then decoded to the original high-dimensional state space. The method is evaluated on simulated cardiac electrodynamics on a 3D ventricular mesh under three stimulation protocols, with comparisons to LSTM, DMD, pure Koopman, and VAR, as well as ablation studies. The central claims are that TK-GCN 'consistently delivers superior predictive accuracy' and that removing any of the three components (Koopman, GCN, Transformer) causes a 'significant drop' in accuracy.

Significance. If the central claims were fully supported, TK-GCN would be a useful integration of three well-known ideas---Koopman linearization, graph convolution, and attention---for a practically relevant problem (cardiac dynamics forecasting). The use of a physiologically grounded simulator and three distinct activation protocols is a sensible testbed, and the paper does provide architectural details and a data availability statement. However, the quantitative evidence in the manuscript does not support the strength of the claimed superiority: the ablation table contradicts the 'consistently superior' and 'any component causes a drop' assertions, and no uncertainty quantification is reported anywhere. The contribution is therefore currently more of an interesting architectural proposal than a substantiated empirical claim.

major comments (3)
  1. [§4.4 (Table 1) and Abstract/Conclusions] The paper's headline claims are contradicted by its own Table 1. Under Protocol II, Transformer + GCN (i.e., without the Koopman operator) achieves lower MSE than TK-GCN in the [0,100) interval (0.00032 vs 0.00037) and in the [100,200) interval (0.00025 vs 0.00051). Under Protocol I, Transformer Only achieves lower MSE than TK-GCN in the [0,100) interval (0.00004 vs 0.00012). These cells directly undermine the abstract's 'consistently delivers superior predictive accuracy' and the conclusion's 'removing any of the three components, i.e., Koopman, GCN, or Transformer, causes a significant drop in prediction accuracy.' The text in §4.4 acknowledges some of these wins (e.g., the Transformer Only result in [0,100) under Protocol I) but does not reconcile them with the strong universal claims, and it omits the Protocol II [100,200) reversal entirely.
  2. [§4.3 and §4.4 (Table 1, Fig. 8)] No uncertainty quantification is provided for any quantitative result. Table 1 reports single MSE values per cell with no number of seeds, no standard deviations, and no significance tests. Figure 8 shows box plots, but the manuscript does not state how many replicate runs each box is based on, nor does it report any statistical test comparing medians. Consequently, differences such as TK-GCN's 0.00044 vs Transformer + CNN-AE's 0.00047 under Protocol II in [200,300) are within typical seed-to-seed variation for deep graph models, and the repeated claims of 'significant' superiority or component contributions are unsupported.
  3. [§4.2 and §5 (baseline description)] The 'Pure Koopman Operator' baseline is not an independent benchmark: it uses the same pre-trained K-GCN encoder and the same learned Koopman matrix K that were trained as part of TK-GCN. Because L_dyn in Eq. (12) explicitly trains K to satisfy K^Δt ψ_e(x(t)) ≈ ψ_e(x(t+Δt)), this baseline largely measures the internal linear-dynamics accuracy of the proposed model rather than comparing against an independently fitted Koopman method. This is acceptable as an ablation of the Transformer component, but presenting it as a state-of-the-art baseline overstates the comparison. Please refit a Koopman-based model (e.g., via EDMD on the latent or original space) or clearly separate the roles of 'ablation' and 'independent baseline' in the text.
minor comments (5)
  1. [§3.1.1 (after Eq. 3)] The phrase 'P denotes the set of Cartesian product of the basis' is grammatically and mathematically unclear; it should read 'the Cartesian product of the sets of basis functions.'
  2. [§3.2.1 (Eq. 17 and surrounding text)] The causal mask is defined as C_ij = 1 for j < i and 0 for j ≥ i, which forbids each position from attending to itself. In the standard autoregressive Transformer decoder for next-token prediction, the mask should allow j ≤ i; as written, the model would be unable to use the current input latent to predict the immediate next latent. Please clarify whether the implementation follows the described mask.
  3. [§4.2] The Aliev–Panfilov model parameters (a, e0, μ1, μ2, k0) are said to be 'adopted from (Aliev and Panfilov, 1996)' but are not listed; please provide the exact parameter values for reproducibility.
  4. [§4.1] The Koopman matrix K is 2496×2496, but no information is given about how K is initialized, whether it is constrained (e.g., to be close to orthogonal), or how λ1 and λ2 in Eq. (14) were chosen. Adding this information would improve reproducibility.
  5. [§4.3 (Fig. 8)] The box plots would benefit from a caption stating the number of test samples and the number of independent model runs per box; currently the y-axis is a symlog scale with a threshold of 0.01, and it is unclear how the threshold applies to the reported values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the central forecasting claim is benchmarked against external baselines and the Koopman component is transparently trained rather than imported as an independent result.

full rationale

The paper's derivation chain is self-contained in the relevant sense. K-GCN is trained as an autoencoder with a learnable Koopman matrix K optimized by Eq. (14); the Transformer is then trained on the frozen latent sequences via Eq. (21); final predictions are decoded and compared against simulated Aliev-Panfilov data. The 'Pure Koopman' baseline explicitly reuses the trained K-GCN encoder and the same learned K, so it functions as an ablation of the Transformer stage rather than an independent first-principles prediction. The paper's claimed contribution is the full TK-GCN pipeline, and its headline evidence is an empirical comparison against external baselines (LSTM, DMD, VAR), which does not reduce to the model's own training objective. The many self-citations (e.g., Wang et al., 2024; Xie and Yao, 2022) are contextual references about cardiac modeling and sequence architectures, not load-bearing mathematical premises or uniqueness theorems. The internal tension between the abstract's 'consistently delivers superior predictive accuracy' and Table 1's short-horizon results (e.g., Transformer Only 0.00004 vs TK-GCN 0.00012 under Protocol I) is a correctness/robustness concern, not a circularity. No equation or defined quantity is equivalent by construction to the result it is used to support.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central empirical claim rests on the trained K matrix and the loss weights lambda1 and lambda2. The Koopman linearization is assumed to hold over long horizons, and the benchmark data come from a single synthetic AP simulation. The pure Koopman baseline uses the paper's own trained operator, so part of the evaluation is self-referential.

free parameters (3)
  • loss weights lambda1 and lambda2 = not reported
    Balance reconstruction, Koopman dynamics, and decay losses in Eq. (14). Values are chosen by hand and not given in the text.
  • sliding window length L = 128
    Fixed in experiments; determines the context available to the Transformer decoder.
  • latent dimension dz = 2496
    Determined by the encoder architecture (two pooling levels). The paper assumes this dimension is sufficient for a near-linear Koopman embedding.
assumptions (4)
  • domain assumption A finite-dimensional matrix K can approximately capture the latent-space Koopman evolution over long horizons.
    Eq. (10) assumes K z(t) is close to z(t+1) for a fixed learned K; the success of this linearization over hundreds of steps is assumed rather than derived.
  • domain assumption The Aliev-Panfilov two-variable model is an adequate surrogate for cardiac electrodynamics for benchmarking.
    All data are generated by this simulation; the paper treats it as a faithful stand-in for real cardiac tissue.
  • domain assumption SplineCNN with B-spline kernels and Graclus pooling preserves the geometric information needed for forecasting.
    Adopted from prior work (Fey et al. 2018; Dhillon et al. 2007); the paper relies on it without additional validation.
  • domain assumption Masked self-attention in the Transformer decoder is sufficient for long-range temporal dependencies in the latent space.
    Standard transformer machinery is used without modification; the paper provides no analysis of whether attention can correct residual nonlinearities left by the Koopman linearization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting." pith.science (2026). https://pith.science/paper/FFQSEGAN

@misc{pith2026250703855,
  author       = {Pith},
  title        = {Pith review of: Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFQSEGAN}},
  note         = {Machine review of arXiv:2507.03855}
}
read the original abstract

Spatiotemporal dynamics forecasting is inherently challenging, particularly in systems defined over irregular geometric domains, due to the need to jointly capture complex spatial correlations and nonlinear temporal dynamics. To tackle these challenges, we propose TK-GCN, a two-stage framework that integrates geometry-aware spatial encoding with long-range temporal modeling. In the first stage, a Koopman-enhanced Graph Convolutional Network (K-GCN) is developed to embed the high-dimensional dynamics distributed on spatially irregular domains into a latent space where the evolution of system states is approximately linear. By leveraging Koopman operator theory, this stage enhances the temporal consistency during the latent learning. In the second stage, a Transformer module is employed to model the temporal progression within the Koopman-encoded latent space. Through the self-attention mechanism, the Transformer captures long-range temporal dependencies, enabling accurate forecasting over extended horizons. We evaluate TK-GCN in spatiotemporal cardiac dynamics forecasting and benchmark its performance against several state-of-the-art baselines. Experimental results and ablation studies show that TK-GCN consistently delivers superior predictive accuracy across a range of forecast horizons, demonstrating its capability to effectively model complex spatial structures and nonlinear temporal dynamics.

Figures

Figures reproduced from arXiv: 2507.03855 by the authors.

Figure 1
Figure 1. Flowchart of the proposed two-stage TK-GCN framework: In the first stage, we propose [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The first-stage K-GCN model to learn a low-dimensional latent representation. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The Transformer decoder architecture for spatiotemporal dynamics forecasting. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The architecture detail of the proposed TK-GCN framework [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Predicted spatiotemporal evolution of electrodynamics under Protocol I at selected time [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Predicted spatiotemporal evolution of electrodynamics under Protocol II at selected [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Predicted spatiotemporal evolution of electrodynamics under Protocol III at selected [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Box plots of MSE for TK-GCN and benchmarks at different predictive time intervals [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

65 extracted references · 52 canonical work pages

  1. [1]

    Aliev, R. R. and Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals , 7(3):293--301

    1996

  2. [2]

    and Joseph, V

    Ba, S. and Joseph, V. R. (2012). Composite gaussian process models for emulating expensive functions. The Annals of Applied Statistics , pages 1838--1860

    work page 2012

  3. [3]

    Biehler, M., Lin, D., and Shi, J. (2024). Detonate: Nonlinear d ynamic e volution modeling of t ime-dependent 3-dimensi ona l poin t cloud profil es. IISE Transactions , 56(5):541--558

    work page 2024

  4. [4]

    M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P

    Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P. (2017). Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine , 34(4):18--42

    work page 2017

  5. [5]

    E., and Deisenroth, M

    Calandra, R., Peters, J., Rasmussen, C. E., and Deisenroth, M. P. (2016). Manifold gaussian processes for regression. In 2016 International joint conference on neural networks (IJCNN) , pages 3338--3345. IEEE

    work page 2016

  6. [6]

    Clevert, D.-A., Unterthiner, T., and Hochreiter, S. (2015). Fast and accurate deep network learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289

    arXiv 2015

  7. [7]

    O., and Gelfand, A

    Datta, A., Banerjee, S., Finley, A. O., and Gelfand, A. E. (2016). Hierarchical nearest-neighbor gaussian process models for large geostatistical datasets. Journal of the American Statistical Association , 111(514):800--812

    work page 2016

  8. [8]

    S., Guan, Y., and Kulis, B

    Dhillon, I. S., Guan, Y., and Kulis, B. (2007). Weighted graph cuts without eigenvectors a multilevel approach. IEEE transactions on pattern analysis and machine intelligence , 29(11):1944--1957

    work page 2007

Show all 65 references
  1. [9]

    E., Weichert, F., and M \"u ller, H

    Fey, M., Lenssen, J. E., Weichert, F., and M \"u ller, H. (2018). Splinecnn: Fast geometric deep learning with continuous b-spline kernels. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 869--877

    2018

  2. [10]

    and Li, W

    Gao, X. and Li, W. (2021). A graph-based lstm model for pm2. 5 forecasting. Atmospheric Pollution Research , 12(9):101150

    2021

  3. [11]

    and Zabaras, N

    Geneva, N. and Zabaras, N. (2022). Transformers for modeling physical systems. Neural Networks , 146:272--289

    2022

  4. [12]

    Geng, X., Li, Y., Wang, L., Zhang, L., Yang, Q., Ye, J., and Liu, Y. (2019). Spatiotemporal multi-graph convolution network for ride-hailing demand forecasting. In Proceedings of the AAAI conference on artificial intelligence , volume 33, pages 3656--3663

    2019

  5. [13]

    L., Amaral, L

    Goldberger, A. L., Amaral, L. A., Glass, L., Hausdorff, J. M., Ivanov, P. C., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C.-K., and Stanley, H. E. (2000). Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. circ...

    2000

  6. [14]

    Gr \"a ler, B., Pebesma, E., and Heuvelink, G. (2016). Spatio-temporal interpolation using gstat. The R Journal

    2016

  7. [15]

    Guo, S., Lin, Y., Feng, N., Song, C., and Wan, H. (2019). Attention based spatial-temporal graph convolutional networks for traffic flow forecasting. In Proceedings of the AAAI conference on artificial intelligence , volume 33, pages 922--929

    2019

  8. [16]

    He, K., Zhang, X., Ren, S., and Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 770--778

    2016

  9. [17]

    G., Filippone, M., and Ghahramani, Z

    Hensman, J., Matthews, A. G., Filippone, M., and Ghahramani, Z. (2015). Mcmc for variationally sparse gaussian processes. Advances in neural information processing systems , 28

    2015

  10. [18]

    Jiang, Y., Fan, J., Liu, Y., and Zhang, X. (2022). Deep graph gaussian processes for short-term traffic flow forecasting from spatiotemporal data. IEEE Transactions on Intelligent Transportation Systems , 23(11):20177--20186

    2022

  11. [19]

    Y., Wen, Q., Zambon, D., Alippi, C., Webb, G

    Jin, M., Koh, H. Y., Wen, Q., Zambon, D., Alippi, C., Webb, G. I., King, I., and Pan, S. (2024). A survey on graph neural networks for time series: Forecasting, classification, imputation, and anomaly detection. IEEE Transactions on Pattern Analysis and Machine Intelligence

    2024

  12. [20]

    C., Lee, S., and Lee, S

    Kwak, J., Ku, D., Jo, J., Wong, S. C., Lee, S., and Lee, S. (2025). A hybrid clustering--regression approach for predicting passenger congestion in a carriage at a subway platform. Expert Systems with Applications , 268:126169

    2025

  13. [21]

    Li, C., Zhang, H., Wang, Z., Wu, Y., and Yang, F. (2022). Spatial-temporal attention mechanism and graph convolutional networks for destination prediction. Frontiers in neurorobotics , 16:925210

    2022

  14. [22]

    Li, Y., Yu, R., Shahabi, C., and Liu, Y. (2017). Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. arXiv preprint arXiv:1707.01926

    2017 arXiv

  15. [23]

    Liu, X., Yeo, K., and Kalagnanam, J. (2018). A statistical modeling approach for spatio-temporal degradation data. Journal of Quality Technology , 50(2):166--182

    2018

  16. [24]

    Liu, X., Yeo, K., and Lu, S. (2022). Statistical modeling for spatio-temporal data from stochastic convection-diffusion processes. Journal of the American Statistical Association , 117(539):1482--1499

    2022

  17. [25]

    Majidizadeh, A., Hasani, H., and Jafari, M. (2024). Semantic segmentation of oblique uav video based on convlstm in complex urban area. Earth Science Informatics , 17(4):3413--3435

    2024

  18. [26]

    Nag, P., Sun, Y., and Reich, B. J. (2023). Spatio-temporal deepkriging for interpolation and probabilistic forecasting. Spatial Statistics , 57:100773

    2023

  19. [27]

    Pascanu, R., Mikolov, T., and Bengio, Y. (2013). On the difficulty of training recurrent neural networks. In International conference on machine learning , pages 1310--1318. Pmlr

    2013

  20. [28]

    Radford, A., Narasimhan, K., Salimans, T., Sutskever, I., et al. (2018). Improving language understanding by generative pre-training. OpenAI blog

    2018

  21. [29]

    Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., Sutskever, I., et al. (2019). Language models are unsupervised multitask learners. OpenAI blog , 1(8):9

    2019

  22. [30]

    Rasmussen, C. E. (2003). Gaussian processes in machine learning. In Summer school on machine learning , pages 63--71. Springer

    2003

  23. [31]

    Senanayake, R., O'callaghan, S., and Ramos, F. (2016). Predicting spatio-temporal propagation of seasonal influenza using variational gaussian process regression. In Proceedings of the AAAI conference on artificial intelligence , volume 30

    2016

  24. [32]

    Shao, W., Jin, Z., Wang, S., Kang, Y., Xiao, X., Menouar, H., Zhang, Z., Zhang, J., and Salim, F. (2022). Long-term spatio-temporal forecasting via dynamic multiple-graph attention. arXiv preprint arXiv:2204.11008

    2022 arXiv

  25. [33]

    Shi, X., Chen, Z., Wang, H., Yeung, D.-Y., Wong, W.-K., and Woo, W.-c. (2015). Convolutional lstm network: A machine learning approach for precipitation nowcasting. Advances in neural information processing systems , 28

    2015

  26. [34]

    D., and Kaiser, W

    Singh, A., Ramos, F., Whyte, H. D., and Kaiser, W. J. (2010). Modeling and decision making in spatio-temporal processes for environmental surveillance. In 2010 IEEE International Conference on Robotics and Automation , pages 5490--5497. IEEE

    2010

  27. [35]

    Snepvangers, J., Heuvelink, G., and Huisman, J. (2003). Soil water content interpolation using spatio-temporal kriging with external drift. Geoderma , 112(3-4):253--271

    2003

  28. [36]

    Tang, T., Mak, S., and Dunson, D. (2024). Hierarchical shrinkage gaussian processes: applications to computer code emulation and dynamical system recovery. SIAM/ASA Journal on Uncertainty Quantification , 12(4):1085--1112

    2024

  29. [37]

    N., Kaiser, ., and Polosukhin, I

    Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, ., and Polosukhin, I. (2017). Attention is all you need. Advances in neural information processing systems , 30

    2017

  30. [38]

    Wang, D., Liu, K., and Zhang, X. (2019). Modeling of a three-dimensional dynamic thermal field under grid-based sensor networks in grain storage. IISE Transactions , 51(5):531--546

    2019

  31. [39]

    Wang, S., Cao, J., and Philip, S. Y. (2020a). Deep learning for spatio-temporal data mining: A survey. IEEE transactions on knowledge and data engineering , 34(8):3681--3700

    2020

  32. [40]

    Wang, T., Chen, K., Lin, W., See, J., Zhang, Z., Xu, Q., and Jia, X. (2020b). Spatio-temporal point process for multiple object tracking. IEEE transactions on neural networks and learning systems , 34(4):1777--1788

    2020

  33. [41]

    Wang, Z., Liu, T., and Yao, B. (2024). Muse-net: Missingness-aware multi-branching self-attention encoder for irregular longitudinal electronic health records. arXiv preprint arXiv:2407.00840

    2024 arXiv

  34. [42]

    and Yao, B

    Wang, Z. and Yao, B. (2021). Multi-branching temporal convolutional network for sepsis prediction. IEEE journal of biomedical and health informatics , 26(2):876--887

    2021

  35. [43]

    Wen, Q., Zhou, T., Zhang, C., Chen, W., Ma, Z., Yan, J., and Sun, L. (2022). Transformers in time series: A survey. arXiv preprint arXiv:2202.07125

    2022 arXiv

  36. [44]

    Wikle, C. K. and Zammit-Mangion, A. (2023). Statistical deep learning for spatial and spatiotemporal data. Annual Review of Statistics and Its Application , 10(1):247--270

    2023

  37. [45]

    Wit, A. L. (1990). Cellular electrophysiologic mechanisms of cardiac arrhythmias. Cardiology Clinics , 8(3):393--410

    1990

  38. [46]

    Wolf, T., Debut, L., Sanh, V., Chaumond, J., Delangue, C., Moi, A., Cistac, P., Rault, T., Louf, R., Funtowicz, M., et al. (2020). Transformers: State-of-the-art natural language processing. In Proceedings of the 2020 conference on empirical methods in natural language process...

    2020

  39. [47]

    Xie, J., Stavrakis, S., and Yao, B. (2024). Automated identification of atrial fibrillation from single-lead ecgs using multi-branching resnet. Frontiers in Physiology , 15:1362185

    2024

  40. [48]

    and Yao, B

    Xie, J. and Yao, B. (2022). Physics-constrained deep active learning for spatiotemporal modeling of cardiac electrodynamics. Computers in Biology and Medicine , 146:105586

    2022

  41. [49]

    and Yao, B

    Xie, J. and Yao, B. (2024). Hierarchical active learning for defect localization in 3d systems. IISE Transactions on Healthcare Systems Engineering , 14(2):115--129

    2024

  42. [50]

    Xu, L., Chen, N., Chen, Z., Zhang, C., and Yu, H. (2021). Spatiotemporal forecasting in earth system science: Methods, uncertainties, predictability and future directions. Earth-Science Reviews , 222:103828

    2021

  43. [51]

    Xu, M., Dai, W., Liu, C., Gao, X., Lin, W., Qi, G.-J., and Xiong, H. (2020). Spatial-temporal transformer networks for traffic flow forecasting. arXiv preprint arXiv:2001.02908

    2020 arXiv

  44. [52]

    Yan, B., Peng, H., Fu, J., Wang, D., and Lu, H. (2021). Learning spatio-temporal transformer for visual tracking. In Proceedings of the IEEE/CVF international conference on computer vision , pages 10448--10457

    2021

  45. [53]

    Yan, H., Paynabar, K., and Shi, J. (2018). Real-time monitoring of high-dimensional functional data streams via spatio-temporal smooth sparse decomposition. Technometrics , 60(2):181--197

    2018

  46. [54]

    D., Liu, X., Pu, L., Chin, S.-m., and Hwang, H.-l

    Yang, H., Yang, J., Han, L. D., Liu, X., Pu, L., Chin, S.-m., and Hwang, H.-l. (2018). A kriging based spatiotemporal approach for traffic volume data imputation. PloS one , 13(4):e0195957

    2018

  47. [55]

    and Yao, B

    Yang, H. and Yao, B. (2023). Sensing, modeling and optimization of cardiac systems: A new generation of digital twin for heart health informatics . Springer Nature

    2023

  48. [56]

    Yao, B. (2021). Spatiotemporal modeling and optimization for personalized cardiac simulation. IISE Transactions on Healthcare Systems Engineering , 11(2):145--160

    2021

  49. [57]

    Yao, B., Leonelli, F., and Yang, H. (2025). Simulation optimization of spatiotemporal dynamics in 3d geometries. IEEE Transactions on Automation Science and Engineering

    2025

  50. [58]

    and Yang, H

    Yao, B. and Yang, H. (2016). Physics-driven spatiotemporal regularization for high-dimensional predictive modeling: A novel approach to solve the inverse ecg problem. Scientific reports , 6(1):39012

    2016

  51. [59]

    Yeung, E., Kundu, S., and Hodas, N. (2019). Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC) , pages 4832--4839. IEEE

    2019

  52. [60]

    Yu, B., Yin, H., and Zhu, Z. (2017). Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. arXiv preprint arXiv:1709.04875

    2017 arXiv

  53. [61]

    and Yao, B

    Zhang, X. and Yao, B. (2025). Geometry-aware active learning of spatiotemporal dynamic systems. arXiv preprint arXiv:2504.19012

    2025 arXiv

  54. [62]

    and Del Castillo, E

    Zhao, X. and Del Castillo, E. (2021). An intrinsic geometrical approach for statistical process control of surface and manifold data. Technometrics , 63(3):295--312

    2021

  55. [63]

    Zheng, C., Fan, X., Wang, C., and Qi, J. (2020). Gman: A graph multi-attention network for traffic prediction. In Proceedings of the AAAI conference on artificial intelligence , volume 34, pages 1234--1241

    2020

  56. [64]

    Zhou, H., Zhang, S., Peng, J., Zhang, S., Li, J., Xiong, H., and Zhang, W. (2021). Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence , volume 35, pages 11106--11115

    2021

  57. [65]

    Zhou, S., Li, J., Wang, H., Shang, S., and Han, P. (2023). Grlstm: trajectory similarity computation with graph-based residual lstm. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 37, pages 4972--4980

    2023

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.