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Interplay between depth and width for interpolation in neural ODEs.

Antonio Álvarez-López1, Arselane Hadj Slimane2, Enrique Zuazua3

  • 1Universidad Autónoma de Madrid, Departamento de Matemáticas, C. Francisco Tomás y Valiente, 7, Madrid, 28049, Spain; Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair for Dynamics, Control, Machine Learning, and Numerics (Alexander von Humboldt Professorship), Cauerstraße, 11, Erlangen, 91058, Germany.

Neural Networks : the Official Journal of the International Neural Network Society
|August 24, 2024
PubMed
Summary

Neural ordinary differential equations (NODEs) offer a control-based approach to supervised learning. This study reveals a trade-off between network width and depth for data interpolation and measure approximation, with implications for autonomous systems.

Keywords:
DepthNeural ODEsSimultaneous controllabilityTransport controlWasserstein distanceWidth

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Area of Science:

  • Machine Learning
  • Dynamical Systems
  • Control Theory

Background:

  • Neural Ordinary Differential Equations (NODEs) are increasingly used for supervised learning, drawing parallels to control theory.
  • The architectural impact of NODEs, specifically width (p) and depth (L), on their learning capabilities requires further elucidation.

Purpose of the Study:

  • To investigate the relationship between network width (p) and depth (L) in Neural Ordinary Differential Equations (NODEs).
  • To analyze how these architectural parameters affect the interpolation of finite datasets and probability measures.
  • To explore data interpolation within the autonomous regime (L=0) of NODEs.

Main Methods:

  • Constructing explicit controls for interpolating finite datasets (D) and probability measures.
  • Analyzing the scaling of depth (L) with respect to width (p) and dataset size (N) or error margin (ɛ).
  • Developing strategies for data interpolation in the autonomous regime (L=0) using probabilistic control and relaxed conditions.

Main Results:

  • A trade-off exists between width (p) and depth (L): L scales as 1+O(N/p) for data interpolation and 1+Op⁻¹+(1+p)⁻¹ɛ⁻ᵈ for measure interpolation.
  • In high-dimensional, wide settings (d,p>N), depth L=0 is achievable.
  • For L=0, probabilistic control strategies (when p=N) and explicit error decay rates (via universal approximation theorems) are established.

Conclusions:

  • Network architecture (width and depth) critically influences NODE performance in supervised learning tasks.
  • The findings provide insights into designing efficient NODE architectures for specific interpolation challenges.
  • The study advances understanding of NODE capabilities, particularly in the autonomous regime, paving the way for novel applications.