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A Y-connected synchronous generator, grounded through a neutral impedance, is designed to produce balanced internal phase voltages with only positive-sequence components. The generator's sequence networks include a source voltage that is exclusively in the positive-sequence network. The sequence components of line-to-ground voltages at the generator terminals illustrate this configuration.
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Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces.

Giovanni S Alberti1, Matteo Santacesaria1, Silvia Sciutto1

  • 1MaLGa Center, Department of Mathematics, University of Genoa , Genova , Italy.

Numerical Functional Analysis and Optimization
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Summary
This summary is machine-generated.

Continuous Generative Neural Networks (CGNNs) model infinite-dimensional functions. This research introduces CGNNs, offering new theoretical guarantees and applications for solving complex inverse problems like signal deblurring.

Keywords:
Generative modelsinjective networksinverse problemsmulti-resolution analysisneural networksvariational autoencoderswavelets

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Area of Science:

  • Machine Learning
  • Functional Analysis
  • Applied Mathematics

Background:

  • Generative models typically operate in finite-dimensional spaces.
  • Infinite-dimensional function spaces present unique modeling challenges.
  • Existing methods struggle with the complexity of continuous data generation.

Purpose of the Study:

  • Introduce Continuous Generative Neural Networks (CGNNs) for infinite-dimensional function spaces.
  • Establish theoretical conditions for CGNN injectivity.
  • Develop applications for inverse problems using CGNNs.

Main Methods:

  • Architecture inspired by DCGAN, adapted for continuous settings using wavelet multiresolution analysis.
  • Analysis of convolutional filters and nonlinear activation functions.
  • Derivation of Lipschitz stability estimates for inverse problems.

Main Results:

  • Conditions for guaranteeing injectivity in CGNNs are presented.
  • CGNNs enable Lipschitz stability estimates for infinite-dimensional inverse problems.
  • Numerical simulations, including signal deblurring, validate the approach.

Conclusions:

  • CGNNs provide a robust framework for generative modeling in continuous, infinite-dimensional spaces.
  • The theoretical framework supports the application of CGNNs to challenging inverse problems.
  • This work opens new avenues for research in generative AI and applied mathematics.