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Grant support

This work was done while the authors were funded by the ERC-DyCon project (grant 694126-DyCon by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme). The work of the first author was funded by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/T024429/1. The work of the second author was funded by the Alexander von Humboldt-Professorship program, the ModConFlex Marie Curie Action, HORIZON-MSCA-2021-DN-01, the COST Action MAT-DYN-NET, the Transregio 154 Project "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks"" of the DFG, grants PID2020-112617GB-C22 and TED2021-131390B-I00 of MINECO (Spain), and by the Madrid Goverment -UAM Agreement for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

Analysis of institutional authors

Zuazua, EnriqueAuthor

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October 30, 2023
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Neural ODE control for classification, approximation, and transport

Publicated to:SIAM REVIEW. 65 (3): 735-773 - 2023-01-01 65(3), DOI: 10.1137/21M1411433

Authors: Ruiz-Balet, D.; Zuazua, E.

Affiliations

Fdn Deusto, Chair Computat Math, Ave Univ 24, Bilbao 48007, Basque Country, Spain - Author
Friedrich Alexander Univ Erlangen Nurnberg, Chair Dynam Control Machine Learning & Numer, Dept Math, D-91058 Erlangen, Germany - Author
Imperial Coll, Dept Math, London SW7 2AZ, England - Author
Univ Autonoma Madrid, Dept Matemat, Madrid 28049, Spain - Author

Abstract

We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the Euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows us to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows us to achieve our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.

Keywords

BoundsContinuity equationData classificationDeep learningExact controllabilityMultilayer feedforward networksNeural odesSimultaneous controlStabilityTimeTotal variation minimizationTransport equationsUniversal approximationWasserstein distance

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal SIAM REVIEW due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2023, it was in position 1/332, thus managing to position itself as a Q1 (Primer Cuartil), in the category Mathematics, Applied. Notably, the journal is positioned above the 90th percentile.

From a relative perspective, and based on the normalized impact indicator calculated from the Field Citation Ratio (FCR) of the Dimensions source, it yields a value of: 34.08, which indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: Dimensions Jul 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-07-19, the following number of citations:

  • WoS: 1
  • Scopus: 17

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-07-19:

  • The use, from an academic perspective evidenced by the Altmetric agency indicator referring to aggregations made by the personal bibliographic manager Mendeley, gives us a total of: 27.
  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 30 (PlumX).

With a more dissemination-oriented intent and targeting more general audiences, we can observe other more global scores such as:

  • The Total Score from Altmetric: 3.
  • The number of mentions on Wikipedia: 3 (Altmetric).

It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

  • The work has been submitted to a journal whose editorial policy allows open Open Access publication.

Leadership analysis of institutional authors

This work has been carried out with international collaboration, specifically with researchers from: Germany; United Kingdom.

There is a significant leadership presence as some of the institution’s authors appear as the first or last signer, detailed as follows: First Author (Ruiz-Balet, Domenec) and Last Author (ZUAZUA IRIONDO, ENRIQUE).

the author responsible for correspondence tasks has been Ruiz-Balet, Domenec.