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A Constructive Approach to Function Realization by Neural Stochastic Differential Equations

arXiv 2307.00215 ↗First posted 1 Jul 2023math.OC cs.LGcited by 0

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Abstract (arXiv)

The problem of function approximation by neural dynamical systems has typically been approached in a top-down manner: Any continuous function can be approximated to an arbitrary accuracy by a sufficiently complex model with a given architecture. This can lead to high-complexity controls which are impractical in applications. In this paper, we take the opposite, constructive approach: We impose various structural restrictions on system dynamics and consequently characterize the class of functions that can be realized by such a system. The systems are implemented as a cascade interconnection of a neural stochastic differential equation (Neural SDE), a deterministic dynamical system, and a readout map. Both probabilistic and geometric (Lie-theoretic) methods are used to characterize the classes of functions realized by such systems.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 11 of its 29 references that are in the map. First posted on arXiv 1 Jul 2023.

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References · 11

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2016Error bounds for approximations with deep ReLU networkscited by 574
  3. 2017Stable Architectures for Deep Neural Networkscited by 341
  4. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  5. 2016Benefits of depth in neural networkscited by 250
  6. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 176
  7. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  8. 2019Deep Learning via Dynamical Systems: An Approximation Perspectivecited by 78
  9. 2021Neural ODE control for classification, approximation and transportcited by 57
  10. 2020An Ode to an ODEcited by 17
  11. 2022Nonlinear controllability and function representation by neural stochastic differential equationscited by 3

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