mlsanityData to 7 Oct 2026

Neural ODE Control for Trajectory Approximation of Continuity Equation

arXiv 2205.09241 ↗First posted 18 May 2022math.OC cs.LG cs.SYcited by 8

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Abstract (arXiv)

We consider the controllability problem for the continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by the flow. We show that the controlled continuity equation has very strong controllability properties. Particularly, a given solution of the continuity equation corresponding to a bounded Lipschitz vector field defines a trajectory on the set of probability measures. For this trajectory, we show that there exist piecewise constant training weights for a neural ODE such that the solution of the continuity equation corresponding to the neural ODE is arbitrarily close to it. As a corollary to this result, we establish that the continuity equation of the neural ODE is approximately controllable on the set of compactly supported probability measures that are absolutely continuous with respect to the Lebesgue measure.

Key facts. Cited by 8 papers in the map (data through 7 Oct 2026). Builds on 14 of its 29 references that are in the map. First posted on arXiv 18 May 2022.

Builds on

14 of 29 references.

References · 14

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2017Stable Architectures for Deep Neural Networkscited by 340
  3. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  4. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 168
  5. 2017Maximum Principle Based Algorithms for Deep Learningcited by 77
  6. 2021Neural ODE control for classification, approximation and transportcited by 56
  7. 2021Momentum Residual Neural Networkscited by 36
  8. 2019Deep learning as optimal control problems: models and numerical methodscited by 33
  9. 2018ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuraciescited by 18
  10. 2018Measure-valued spline curves: an optimal transport viewpointcited by 17
  11. 2020Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theorycited by 17
  12. 2019Mean-Field Neural ODEs via Relaxed Optimal Controlcited by 16
2 more
  1. 2015Optimal Control of Continuity Equationscited by 4
  2. 2017Approximate and exact controllability of the continuity equation with a localized vector fieldcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 3; 2025: 1; 2026: 1.

8 papers.

  1. 2024Measure-to-measure interpolation using Transformerscited by 28
  2. 2024Interplay between depth and width for interpolation in neural ODEscited by 8
  3. 2023Double-well Net for Image Segmentationcited by 7
  4. 2023Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEscited by 4
  5. 2024Constructive approximate transport maps with normalizing flowscited by 4
  6. 2025Control, Optimal Transport and Neural Differential Equations in Supervised Learningcited by 1
  7. 2023Noise in the reverse process improves the approximation capabilities of diffusion modelscited by 0
  8. 2026Constructive interpolation and generalization rates for neural ODEs: a control perspectivecited by 0