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arxiv: 2606.19138 · v1 · pith:SUDPUL3Snew · submitted 2026-06-17 · 💻 cs.LG · stat.ML

INDEQS: Informed Neural controlled Differential EQuationS

Michael Detzel , Gabriel Nobis , Kristiyan Blagov , Juri Schubert , Jackie Ma , Wojciech Samek This is my paper

Pith reviewed 2026-06-26 21:02 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords neural controlled differential equationsgraph neural networkstime series forecastingspatio-temporal modelinginformed neural networksadvection simulationriver discharge forecastingtraffic flow prediction
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The pith

Separating inner and outer graph mixing in NCDEs improves forecasting accuracy when a directed graph is known ahead of time.

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

The paper introduces INDEQS to embed known directed-graph structure into neural controlled differential equations at two distinct points rather than learning everything from data. It separates the mixing of hidden states across nodes (inner) from the mixing between the vector field and the control signal (outer), and supplies both a strictly constrained variant and one that learns extra edges via adaptive convolutions. Experiments on a new continuous advection simulator plus real river-discharge and traffic-flow tasks show that outer informedness lowers mean absolute error relative to an uninformed NCDE of similar size, with the gain growing on larger graphs, while inner informedness is more parameter-efficient when the known adjacency must be respected exactly. Continuous decoders further outperform discrete ones on the real datasets. The work therefore supplies both a concrete architectural choice and a synthetic benchmark for testing when prior graph knowledge helps continuous-time forecasting.

Core claim

INDEQS extends the NCDE framework by inserting a known directed graph at two architecturally separate positions: inner mixing that propagates hidden states across nodes according to the adjacency, and outer mixing that modulates the interaction between the learned vector field and the control path. The method supplies a lightweight graph-constrained variant that enforces the known edges and a more expressive variant that augments them with adaptive graph convolutions learned from data. On a continuous advection simulation that generates synthetic spatio-temporal series with known ground-truth flow, as well as on hydrological river networks and the PeMS08 traffic dataset, outer informedness y

What carries the argument

Separation of inner mixing (hidden-state propagation across graph nodes) from outer mixing (vector-field-to-control interaction), allowing the known directed adjacency to be injected at either or both locations.

If this is right

  • Outer informedness lowers mean absolute error relative to an uninformed NCDE of comparable size, with larger gains on bigger graphs.
  • Inner informedness supplies a parameter-efficient option when the supplied adjacency must be followed exactly.
  • Continuous-time decoders produce higher accuracy and greater temporal flexibility than discrete convolutional decoders on real hydrological and traffic tasks.
  • The continuous advection simulation on directed graphs supplies a controlled testbed with known ground-truth flow for studying graph-informed continuous models.

Where Pith is reading between the lines

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

  • The inner/outer split could be tested on other known-topology domains such as power-grid load or epidemic spread on contact networks.
  • Replacing the adaptive graph convolutions with alternative graph-learning layers would isolate whether the benefit comes from the informed positions or from the extra parameters.
  • The advection simulator could be extended to include stochastic forcing or time-varying edges to probe robustness of the informedness gains.

Load-bearing premise

A directed graph structure is known in advance and placing the graph information at the inner-versus-outer mixing split is a structurally useful insertion point.

What would settle it

An uninformed NCDE with the same parameter budget achieves equal or lower MAE than the outer-informed INDEQS variant on the PeMS08 traffic dataset or on the largest advection graphs.

Figures

Figures reproduced from arXiv: 2606.19138 by Gabriel Nobis, Jackie Ma, Juri Schubert, Kristiyan Blagov, Michael Detzel, Wojciech Samek.

Figure 1
Figure 1. Figure 1: Overview: Starting from an edge transition graph with prior known structure, we simulate advec [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Difference in information transport between hidden state [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: NCDE encoder and convolutional decoder. Encoding (left half): Up to time T, the data points at the bottom are interpolated to obtain a continuous path X, whose derivative is then multiplied with f, to obtain dH. A allows now direct mixing – at the outer position – of the nodes’ control dH, depending on dH. Only after occurs the multiplication with g to obtain the final update dZ. A mixing at the inner posi… view at source ↗
Figure 4
Figure 4. Figure 4: Advection of an initial Gaussian pulse on a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Advection on graph edges for a 4-node graph for 3 time steps and increasing node resolution. The additional intermediate nodes are depicted as black dots. The last image additionally shows a zoomed out view of another time step with future masses along the virtual edges (green) where masses enter at the top right edges and leave the graph on the bottom edge. For the experiments on graph advection data, we … view at source ↗
Figure 6
Figure 6. Figure 6: MAE over 5 seed runs over graphs with increasing number of nodes for the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean MAE over 5 runs for increasing spatial node resolution for the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average performance of INDEQS with increasing adjacency matrix powers vs. the uninformed [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Graphical overview of the NCDE encoder-decoder structure in combination with two continuous [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Average performance of INDEQS and additional informedness variations, with increasing adja [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: MAE and runtime for different solver and step-size combinations for the discrete convolutional [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The graph network of the discharge stations of the Weser river and their catchments [PITH_FULL_IMAGE:figures/full_fig_p038_12.png] view at source ↗
read the original abstract

Neural Controlled Differential Equations (NCDE) provide a powerful continuous-time framework for forecasting time series, but standard graph-based extensions typically learn spatial structure purely from data, even in settings where a directed graph structure is known a priori. We introduce Informed Neural controlled Differential EQuationS (INDEQS), a graph-based NCDE forecasting method that incorporates prior knowledge of a directed graph at distinct architectural positions. INDEQS separates inner mixing of hidden states across graph nodes from outer mixing between vector field and control, and offers both a lightweight graph-constrained variant and a more expressive variant, learning additional graph connections from data via adaptive graph convolutions. To systematically study when graph informedness is beneficial in forecasting, we devise a continuous advection simulation on directed graphs, yielding synthetic spatio-temporal datasets with known ground-truth flow structure. We then evaluate INDEQS on two real-world tasks: river discharge forecasting on a hydrological network and traffic flow prediction on PeMS08. Across these synthetic and real-world benchmarks, outer informedness consistently improves mean absolute error over an uninformed NCDE with comparable parameter count, particularly on larger graphs, while inner informedness offers a more parameter-efficient alternative when strict adherence to a known adjacency is desired. A comparison of discrete convolutional and continuous-time decoders further shows that continuous decoders yield better accuracy and greater temporal flexibility on real-world tasks. An implementation of INDEQS and the advection simulation is available at https://github.com/Mitchi1/indeqs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes INDEQS, an extension of Neural Controlled Differential Equations (NCDEs) for spatio-temporal forecasting that injects a priori directed graph structure at two distinct positions: inner mixing of hidden states across nodes and outer mixing between the vector field and control path. It introduces a lightweight graph-constrained variant and an expressive variant using adaptive graph convolutions to learn additional edges. A new synthetic continuous advection simulation on directed graphs is presented to generate datasets with known ground-truth flow. Evaluations on river discharge forecasting and PeMS08 traffic prediction claim that outer informedness yields consistent MAE gains over an uninformed NCDE with matched parameter count (especially on larger graphs), while inner informedness is more parameter-efficient when strict adherence to the known adjacency is required. A decoder comparison shows continuous-time decoders outperform discrete convolutional ones on real tasks. Code and the advection simulator are released.

Significance. If the reported MAE improvements hold under fair hyperparameter matching and are specifically attributable to the inner/outer injection points rather than graph information in general, the work offers a practical way to incorporate known directed graph priors into continuous-time models. The synthetic advection generator and open implementation are clear strengths that support reproducibility and further investigation of graph-informed NCDEs.

major comments (2)
  1. [Method description and Experiments] The central architectural claim is that separating inner hidden-state mixing from outer vector-field/control mixing provides a structurally useful insertion point for known directed adjacency (as opposed to other positions). However, the experiments section reports comparisons only against an uninformed NCDE baseline and does not include ablations that inject the identical adjacency matrix at alternative locations (e.g., initial hidden-state projection, control-path preprocessing, or decoder). Without these controls it is impossible to determine whether the observed MAE reductions on the advection, river, and traffic benchmarks arise from the inner/outer split specifically or from the mere presence of graph information. This directly affects the load-bearing claim that the proposed separation is the advantageous design choice.
  2. [Experiments] The abstract and experimental claims state that outer informedness 'consistently improves mean absolute error' and that gains are 'particularly on larger graphs,' yet no tables or text report statistical significance tests, standard deviations across random seeds, or explicit confirmation that all baselines received equivalent hyperparameter tuning and training budgets. Given that the soundness assessment notes uncertainty about baseline fairness, these details are required to substantiate the empirical contribution.
minor comments (1)
  1. [Method] Notation for the inner and outer mixing operators should be introduced with explicit equations early in the method section to avoid ambiguity when the two variants are later compared.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's comments. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Method description and Experiments] The central architectural claim is that separating inner hidden-state mixing from outer vector-field/control mixing provides a structurally useful insertion point for known directed adjacency (as opposed to other positions). However, the experiments section reports comparisons only against an uninformed NCDE baseline and does not include ablations that inject the identical adjacency matrix at alternative locations (e.g., initial hidden-state projection, control-path preprocessing, or decoder). Without these controls it is impossible to determine whether the observed MAE reductions on the advection, river, and traffic benchmarks arise from the inner/outer split specifically or from the mere presence of graph information. This directly affects the load-bearing claim that the proposed separation is the advantageous design choice.

    Authors: We recognize that the referee's point is valid and that additional ablations would more conclusively attribute the gains to the specific inner/outer positions rather than graph information in general. Our experiments were designed to compare the proposed INDEQS variants against the standard uninformed NCDE with matched parameter counts, demonstrating the benefit of informedness at these positions. The separation is theoretically motivated by the NCDE formulation, where the vector field and hidden state dynamics are distinct. To strengthen the manuscript, we will add a new subsection with ablations injecting the graph at alternative locations (initial hidden state and decoder) on the synthetic advection dataset, as this is computationally feasible. This will be included in the revised version. revision: yes

  2. Referee: [Experiments] The abstract and experimental claims state that outer informedness 'consistently improves mean absolute error' and that gains are 'particularly on larger graphs,' yet no tables or text report statistical significance tests, standard deviations across random seeds, or explicit confirmation that all baselines received equivalent hyperparameter tuning and training budgets. Given that the soundness assessment notes uncertainty about baseline fairness, these details are required to substantiate the empirical contribution.

    Authors: We agree that reporting statistical details is important for substantiating the claims. In the current manuscript, we focused on mean performance but did not include variance or significance tests. We will revise the experimental section to report mean and standard deviation over at least 5 random seeds for all methods, include p-values from statistical tests where appropriate, and add a paragraph detailing the hyperparameter tuning procedure and training budgets to confirm fairness. These additions will be made in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: architectural proposal evaluated empirically without self-referential derivations

full rationale

The paper proposes INDEQS as a graph-informed extension of NCDE by separating inner hidden-state mixing from outer vector-field/control mixing and injecting a known directed adjacency at those positions. Claims of improved MAE are supported solely by direct empirical comparisons on a custom advection simulation plus river/traffic benchmarks, using standard train/test splits and parameter-matched baselines. No equations, uniqueness theorems, or ansatzes are introduced that reduce any result to a fitted quantity defined by the same model or to a self-citation chain. The method is presented as an architectural choice whose value is tested externally rather than derived by construction from its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The method implicitly assumes the existence of a known directed adjacency matrix and that the NCDE vector field can be meaningfully modulated by graph operations at the chosen positions.

pith-pipeline@v0.9.1-grok · 5812 in / 1250 out tokens · 20019 ms · 2026-06-26T21:02:25.151340+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

180 extracted references · 40 canonical work pages

  1. [1]

    Tanyu, Derick Nganyu and Ning, Jianfeng and Freudenberg, Tom and Heilenkötter, Nick and Rademacher, Andreas and Iben, Uwe and Maass, Peter , month = may, year =. Deep

  2. [2]

    Gaussian

    Härkönen, Marc and Lange-Hegermann, Markus and Raiţă, Bogdan , month = jul, year =. Gaussian

  3. [3]

    Anumasa, Srinivas and Gunapati, Geetakrishnasai and Srijith, P. K. , month = dec, year =. Continuous

  4. [4]

    On neural differential equations , volume =

    Kidger, P , editor =. On neural differential equations , volume =. , pages =. 2021 , series =

    2021

  5. [5]

    Ali, Ameen and Schnake, Thomas and Eberle, Oliver and Montavon, Grégoire and Müller, Klaus-Robert and Wolf, Lior , month = jun, year =

  6. [6]

    Universal approximation property of invertible neural networks , url =

    Ishikawa, Isao and Teshima, Takeshi and Tojo, Koichi and Oono, Kenta and Ikeda, Masahiro and Sugiyama, Masashi , month = apr, year =. Universal approximation property of invertible neural networks , url =

  7. [7]

    Training

    Krämer, Andreas and Köhler, Jonas and Noé, Frank , month = nov, year =. Training

  8. [8]

    Stochastic

    Wu, Hao and Köhler, Jonas and Noé, Frank , month = oct, year =. Stochastic

  9. [9]

    Spectral

    Bruna, Joan and Zaremba, Wojciech and Szlam, Arthur and LeCun, Yann , month = may, year =. Spectral

  10. [11]

    and Neumann, David and Samek, Wojciech and Müller, Klaus-Robert and Lapuschkin, Sebastian , month = jun, year =

    Anders, Christopher J. and Neumann, David and Samek, Wojciech and Müller, Klaus-Robert and Lapuschkin, Sebastian , month = jun, year =. Software for

  11. [12]

    Köhler, Jonas and Krämer, Andreas and Noé, Frank , month = nov, year =. Smooth

  12. [13]

    Kipf and Max Welling , booktitle=

    Thomas N. Kipf and Max Welling , booktitle=. 2017 , url=

    2017

  13. [14]

    Kipf and Max Welling , year=

    Thomas N. Kipf and Max Welling , year=. 1609.02907 , archivePrefix=

    Pith/arXiv arXiv

  14. [15]

    and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , month = feb, year =

    Song, Yang and Sohl-Dickstein, Jascha and Kingma, Diederik P. and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , month = feb, year =. Score-

  15. [16]

    Scientific Reports , author =

    Recurrent. Scientific Reports , author =. 2018 , keywords =. doi:10.1038/s41598-018-24271-9 , abstract =

    work page doi:10.1038/s41598-018-24271-9 2018

  16. [17]

    Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em , month = nov, year =. Physics

  17. [18]

    and Nakajima, Shinichi and Kessel, Pan , month = jun, year =

    Vaitl, Lorenz and Nicoli, Kim A. and Nakajima, Shinichi and Kessel, Pan , month = jun, year =. Path-

  18. [19]

    Gao, Jianfei and Ribeiro, Bruno , month = mar, year =. On the

  19. [20]

    Shrikumar, Avanti and Greenside, Peyton and Shcherbina, Anna and Kundaje, Anshul , month = apr, year =. Not

  20. [21]

    Zubov, Kirill and McCarthy, Zoe and Ma, Yingbo and Calisto, Francesco and Pagliarino, Valerio and Azeglio, Simone and Bottero, Luca and Luján, Emmanuel and Sulzer, Valentin and Bharambe, Ashutosh and Vinchhi, Nand and Balakrishnan, Kaushik and Upadhyay, Devesh and Rackauckas, Chris , month = jul, year =

  21. [22]

    Morrill, James and Kidger, Patrick and Yang, Lingyi and Lyons, Terry , month = jun, year =. Neural

  22. [23]

    Kidger, Patrick and Morrill, James and Foster, James and Lyons, Terry , booktitle =

  23. [24]

    and Tancik, Matthew and Barron, Jonathan T

    Mildenhall, Ben and Srinivasan, Pratul P. and Tancik, Matthew and Barron, Jonathan T. and Ramamoorthi, Ravi and Ng, Ren , month = aug, year =

  24. [25]

    Multimodal

    Lahat, Dana and Adali, Tülay and Jutten, Christian , year =. Multimodal. doi:10.1109/JPROC.2015.2460697 , journal =

    work page doi:10.1109/jproc.2015.2460697 2015

  25. [26]

    Sensors , author =

    Multi-. Sensors , author =. 2022 , keywords =. doi:10.3390/s22103697 , abstract =

    work page doi:10.3390/s22103697 2022

  26. [27]

    IEEE Transactions on Intelligent Transportation Systems , author =

    Modeling and. IEEE Transactions on Intelligent Transportation Systems , author =. 2014 , keywords =. doi:10.1109/TITS.2013.2278614 , urldate =

    work page doi:10.1109/tits.2013.2278614 2014

  27. [28]

    Digital Signal Processing , author =

    Methods for interpreting and understanding deep neural networks , volume =. Digital Signal Processing , author =. 2018 , keywords =. doi:10.1016/j.dsp.2017.10.011 , language =

    work page doi:10.1016/j.dsp.2017.10.011 2018

  28. [29]

    Papamakarios, George and Pavlakou, Theo and Murray, Iain , month = jun, year =. Masked

  29. [30]

    Linear algebra with transformers , url =

    Charton, François , month = nov, year =. Linear algebra with transformers , url =

  30. [31]

    Journal of Computational Physics , author =

    Learning. Journal of Computational Physics , author =. 2021 , keywords =. doi:10.1016/j.jcp.2020.110072 , abstract =

    work page doi:10.1016/j.jcp.2020.110072 2021

  31. [32]

    Rubanova, Yulia and Chen, Ricky T. Q. and Duvenaud, David , month = jul, year =. Latent

  32. [33]

    Rubanova, Yulia and Chen, Ricky T. Q. and Duvenaud, David K , booktitle =

  33. [34]

    Langevin

    Srinivasan, Vignesh and Müller, Klaus-Robert and Samek, Wojciech and Nakajima, Shinichi , month = aug, year =. Langevin

  34. [35]

    Lagrangian

    Cranmer, Miles and Greydanus, Sam and Hoyer, Stephan and Battaglia, Peter and Spergel, David and Ho, Shirley , month = jul, year =. Lagrangian

  35. [36]

    and Ying, Rex and Leskovec, Jure , month = sep, year =

    Hamilton, William L. and Ying, Rex and Leskovec, Jure , month = sep, year =. Inductive

  36. [37]

    Hopfield

    Ramsauer, Hubert and Schäfl, Bernhard and Lehner, Johannes and Seidl, Philipp and Widrich, Michael and Adler, Thomas and Gruber, Lukas and Holzleitner, Markus and Pavlović, Milena and Sandve, Geir Kjetil and Greiff, Victor and Kreil, David and Kopp, Michael and Klambauer, Günter and Brandstetter, Johannes and Hochreiter, Sepp , month = apr, year =. Hopfield

  37. [38]

    Gu, Albert and Dao, Tri and Ermon, Stefano and Rudra, Atri and Re, Christopher , month = oct, year =

  38. [39]

    Machine Learning: Science and Technology , author =

    High-fidelity molecular dynamics trajectory reconstruction with bi-directional neural networks , volume =. Machine Learning: Science and Technology , author =. 2022 , keywords =. doi:10.1088/2632-2153/ac6ec6 , abstract =

    work page doi:10.1088/2632-2153/ac6ec6 2022

  39. [40]

    Salinas, David and Bohlke-Schneider, Michael and Callot, Laurent and Medico, Roberto and Gasthaus, Jan , month = oct, year =. High-

  40. [41]

    Raissi, Maziar and Yazdani, Alireza and Karniadakis, George Em , month = aug, year =. Hidden

  41. [42]

    , month = jun, year =

    Liu, Zibo and Shojaee, Parshin and Reddy, Chandan K. , month = jun, year =. Graph-based

  42. [43]

    Poli, Michael and Massaroli, Stefano and Park, Junyoung and Yamashita, Atsushi and Asama, Hajime and Park, Jinkyoo , booktitle =. Graph. 2021 , maintitle =

    2021

  43. [44]

    1911.07532 , archivePrefix=

    Michael Poli and Stefano Massaroli and Junyoung Park and Atsushi Yamashita and Hajime Asama and Jinkyoo Park , year=. 1911.07532 , archivePrefix=

    arXiv 1911

  44. [45]

    Zhou, Jie and Cui, Ganqu and Hu, Shengding and Zhang, Zhengyan and Yang, Cheng and Liu, Zhiyuan and Wang, Lifeng and Li, Changcheng and Sun, Maosong , month = oct, year =. Graph

  45. [46]

    ISPRS International Journal of Geo-Information , author =

    Graph. ISPRS International Journal of Geo-Information , author =. 2023 , pages =. doi:10.3390/ijgi12030100 , abstract =

    work page doi:10.3390/ijgi12030100 2023

  46. [47]

    Jeongwhan Choi AND Hwangyong Choi AND Jeehyun Hwang AND Noseong Park , booktitle=

  47. [48]

    2022 , month=

    Proceedings of the AAAI Conference on Artificial Intelligence , author=. 2022 , month=. doi:10.1609/aaai.v36i6.20587 , abstractNote=

    work page doi:10.1609/aaai.v36i6.20587 2022

  48. [49]

    , month = sep, year =

    Chamberlain, Benjamin Paul and Rowbottom, James and Gorinova, Maria and Webb, Stefan and Rossi, Emanuele and Bronstein, Michael M. , month = sep, year =

  49. [50]

    2021 , editor =

    Chamberlain, Ben and Rowbottom, James and Gorinova, Maria I and Bronstein, Michael and Webb, Stefan and Rossi, Emanuele , booktitle =. 2021 , editor =

    2021

  50. [51]

    Machine Learning: Science and Technology , author =

    Gradients should stay on path: better estimators of the reverse- and forward. Machine Learning: Science and Technology , author =. 2022 , keywords =. doi:10.1088/2632-2153/ac9455 , abstract =

    work page doi:10.1088/2632-2153/ac9455 2022

  51. [52]

    and Bruna, Joan and Cohen, Taco and Veličković, Petar , month = may, year =

    Bronstein, Michael M. and Bruna, Joan and Cohen, Taco and Veličković, Petar , month = may, year =. Geometric

  52. [53]

    Pattern Recognition , author =

    Explaining. Pattern Recognition , author =. 2017 , keywords =. doi:10.1016/j.patcog.2016.11.008 , abstract =

    work page doi:10.1016/j.patcog.2016.11.008 2017

  53. [54]

    Enhancing the

    Li, Shiyang and Jin, Xiaoyong and Xuan, Yao and Zhou, Xiyou and Chen, Wenhu and Wang, Yu-Xiang and Yan, Xifeng , month = jan, year =. Enhancing the

  54. [55]

    Dissipative

    Zhong, Yaofeng Desmond and Dey, Biswadip and Chakraborty, Amit , month = apr, year =. Dissipative

  55. [56]

    Discovering

    Cranmer, Miles and Sanchez-Gonzalez, Alvaro and Battaglia, Peter and Xu, Rui and Cranmer, Kyle and Spergel, David and Ho, Shirley , month = nov, year =. Discovering

  56. [57]

    Aziznejad, Shayan and Unser, Michael , month = may, year =. Deep. doi:10.1109/ICASSP.2019.8682547 , urldate =

    work page doi:10.1109/icassp.2019.8682547 2019

  57. [58]

    Continuous

    Deng, Zhiwei and Nawhal, Megha and Meng, Lili and Mori, Greg , month = sep, year =. Continuous

  58. [59]

    Combining

    Gu, Albert and Johnson, Isys and Goel, Karan and Saab, Khaled and Dao, Tri and Rudra, Atri and Ré, Christopher , month = oct, year =. Combining

  59. [60]

    Zhou, Zhaorun and Shi, Zhenghao and Guo, Mingtao and Feng, Yaning and Zhao, Minghua , month = may, year =

  60. [61]

    Axiomatic

    Sundararajan, Mukund and Taly, Ankur and Yan, Qiqi , month = jun, year =. Axiomatic

  61. [62]

    Augmented

    Dupont, Emilien and Doucet, Arnaud and Teh, Yee Whye , month = oct, year =. Augmented

  62. [63]

    Augmented

    Dupont, Emilien and Doucet, Arnaud and Teh, Yee Whye , booktitle =. Augmented

  63. [64]

    and Kaiser, Lukasz and Polosukhin, Illia , month = dec, year =

    Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N. and Kaiser, Lukasz and Polosukhin, Illia , month = dec, year =. Attention

  64. [65]

    Attention based spatial-temporal graph convolutional networks for traffic flow forecasting,

    Attention. Proceedings of the AAAI Conference on Artificial Intelligence , author =. 2019 , keywords =. doi:10.1609/aaai.v33i01.3301922 , abstract =

    work page doi:10.1609/aaai.v33i01.3301922 2019

  65. [66]

    and Reisinger, Christoph and Wang, Sheng , month = aug, year =

    Cohen, Samuel N. and Reisinger, Christoph and Wang, Sheng , month = aug, year =. Arbitrage-free neural-

  66. [67]

    Neural Computation , author =

    A. Neural Computation , author =. 2020 , keywords =. doi:10.1162/neco_a_01273 , abstract =

    work page doi:10.1162/neco_a_01273 2020

  67. [68]

    Sixt, Leon and Landgraf, Tim , month = nov, year =. A

  68. [69]

    Chevyrev, Ilya and Kormilitzin, Andrey , month = mar, year =. A

  69. [70]

    Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David K , booktitle =

  70. [71]

    Universal

    Rackauckas, Christopher and Ma, Yingbo and Martensen, Julius and Warner, Collin and Zubov, Kirill and Supekar, Rohit and Skinner, Dominic and Ramadhan, Ali and Edelman, Alan , month = nov, year =. Universal

  71. [72]

    , volume =

    Estimation of non-normalized statistical models by score matching. , volume =. Journal of Machine Learning Research , author =. 2005 , file =

    2005

  72. [73]

    Raissi, P

    Physics-informed neural networks:. Journal of Computational Physics , author =. 2019 , keywords =. doi:https://doi.org/10.1016/j.jcp.2018.10.045 , abstract =

    work page doi:10.1016/j.jcp.2018.10.045 2019

  73. [74]

    Differential equations driven by rough paths , publisher =

    Lyons, Terry J and Caruana, Michael and Lévy, Thierry , year =. Differential equations driven by rough paths , publisher =

  74. [75]

    Introduction to

    Samek, Wojciech , file =. Introduction to

  75. [76]

    Ma, Yao and Tang, Jiliang , keywords =. Book

  76. [77]

    Raissi, Maziar , month = jan, year =. Deep

  77. [78]

    Hoidn, Oliver and Mishra, Aashwin Ananda and Mehta, Apurva , month = oct, year =. Physics

  78. [79]

    & Spirtes, P

    Review of. Frontiers in Genetics , author =. 2019 , keywords =. doi:10.3389/fgene.2019.00524 , urldate =

    work page doi:10.3389/fgene.2019.00524 2019

  79. [80]

    Causal discovery for time series from multiple datasets with latent contexts , url =

    Günther, Wiebke and Ninad, Urmi and Runge, Jakob , month = jun, year =. Causal discovery for time series from multiple datasets with latent contexts , url =

  80. [81]

    Journal of Climate , author =

    Quantifying the. Journal of Climate , author =. 2014 , keywords =. doi:10.1175/JCLI-D-13-00159.1 , abstract =

    work page doi:10.1175/jcli-d-13-00159.1 2014

Showing first 80 references.