---
res:
  bibo_abstract:
  - Continuous-time neural networks are a class of machine learning systems that can
    tackle representation learning on spatiotemporal decision-making tasks. These
    models are typically represented by continuous differential equations. However,
    their expressive power when they are deployed on computers is bottlenecked by
    numerical differential equation solvers. This limitation has notably slowed down
    the scaling and understanding of numerous natural physical phenomena such as the
    dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving
    the given dynamical system in closed form. This is known to be intractable in
    general. Here, we show that it is possible to closely approximate the interaction
    between neurons and synapses—the building blocks of natural and artificial neural
    networks—constructed by liquid time-constant networks efficiently in closed form.
    To this end, we compute a tightly bounded approximation of the solution of an
    integral appearing in liquid time-constant dynamics that has had no known closed-form
    solution so far. This closed-form solution impacts the design of continuous-time
    and continuous-depth neural models. For instance, since time appears explicitly
    in closed form, the formulation relaxes the need for complex numerical solvers.
    Consequently, we obtain models that are between one and five orders of magnitude
    faster in training and inference compared with differential equation-based counterparts.
    More importantly, in contrast to ordinary differential equation-based continuous
    networks, closed-form networks can scale remarkably well compared with other deep
    learning instances. Lastly, as these models are derived from liquid networks,
    they show good performance in time-series modelling compared with advanced recurrent
    neural network models.@eng
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Ramin
      foaf_name: Hasani, Ramin
      foaf_surname: Hasani
  - foaf_Person:
      foaf_givenName: Mathias
      foaf_name: Lechner, Mathias
      foaf_surname: Lechner
      foaf_workInfoHomepage: http://www.librecat.org/personId=3DC22916-F248-11E8-B48F-1D18A9856A87
  - foaf_Person:
      foaf_givenName: Alexander
      foaf_name: Amini, Alexander
      foaf_surname: Amini
  - foaf_Person:
      foaf_givenName: Lucas
      foaf_name: Liebenwein, Lucas
      foaf_surname: Liebenwein
  - foaf_Person:
      foaf_givenName: Aaron
      foaf_name: Ray, Aaron
      foaf_surname: Ray
  - foaf_Person:
      foaf_givenName: Max
      foaf_name: Tschaikowski, Max
      foaf_surname: Tschaikowski
  - foaf_Person:
      foaf_givenName: Gerald
      foaf_name: Teschl, Gerald
      foaf_surname: Teschl
  - foaf_Person:
      foaf_givenName: Daniela
      foaf_name: Rus, Daniela
      foaf_surname: Rus
  bibo_doi: 10.1038/s42256-022-00556-7
  bibo_issue: '11'
  bibo_volume: 4
  dct_date: 2022^xs_gYear
  dct_identifier:
  - UT:000884215600003
  dct_isPartOf:
  - http://id.crossref.org/issn/2522-5839
  dct_language: eng
  dct_publisher: Springer Nature@
  dct_subject:
  - Artificial Intelligence
  - Computer Networks and Communications
  - Computer Vision and Pattern Recognition
  - Human-Computer Interaction
  - Software
  dct_title: Closed-form continuous-time neural networks@
...