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Approximation bounds for convolutional neural networks in operator learning.

Nicola Rares Franco1, Stefania Fresca1, Andrea Manzoni1

  • 1MOX, Math Department, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy.

Neural Networks : the Official Journal of the International Neural Network Society
|February 6, 2023
PubMed
Summary
This summary is machine-generated.

This study provides rigorous error bounds for deep Convolutional Neural Networks (CNNs) approximating nonlinear operators. The findings reveal a connection between CNNs and Fourier transforms, offering mathematical foundations for these models.

Keywords:
Approximation theoryConvolutional neural networksOperator learning

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Area of Science:

  • Computational Mathematics
  • Applied Mathematics
  • Machine Learning

Background:

  • Deep Convolutional Neural Networks (CNNs) show promise in reduced order modeling of parametrized partial differential equations (PDEs).
  • Existing CNN approaches for PDEs lack rigorous mathematical justification and error analysis.
  • Understanding the theoretical underpinnings of CNNs is crucial for reliable scientific modeling.

Purpose of the Study:

  • To derive rigorous error bounds for approximating nonlinear operators using CNN models.
  • To provide a mathematical foundation for the use of CNNs in scientific computing.
  • To interpret the impact of CNN hyperparameters on approximation accuracy.

Main Methods:

  • Developing constructive proofs for error estimation in CNN-based operator approximation.
  • Analyzing the mapping of finite-dimensional inputs to functional outputs via CNNs.
  • Investigating the relationship between CNN architectures and the Fourier transform.

Main Results:

  • Rigorous error bounds were derived for CNN approximations of nonlinear operators.
  • The error estimates offer insights into the role of hyperparameters in CNN performance.
  • A significant connection was established between CNNs and Fourier transform properties.
  • Numerical experiments validated the theoretical error bounds.

Conclusions:

  • The study establishes a strong mathematical foundation for employing CNNs in reduced order modeling.
  • The derived error bounds enhance the interpretability and reliability of CNN models in scientific applications.
  • The revealed link to Fourier transforms opens new avenues for CNN design and analysis in PDE modeling.