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Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches
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Two-hidden-layer ReLU neural networks and finite elements.

Pengzhan Jin1

  • 1National Engineering Laboratory for Big Data Analysis and Applications, Peking University, Beijing, 100871, China.

Neural Networks : the Official Journal of the International Neural Network Society
|January 14, 2026
PubMed
Summary

Piecewise linear functions on convex polytope meshes can be weakly represented by two-hidden-layer Rectified Linear Unit (ReLU) neural networks. Neuron counts are precisely determined by mesh complexity, linking neural networks and finite element analysis.

Keywords:
Finite elementsReLU neural networksWeak representation

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Area of Science:

  • Computational Mathematics
  • Artificial Intelligence
  • Numerical Analysis

Background:

  • Piecewise linear functions are fundamental in numerical methods like finite element analysis.
  • Rectified Linear Unit (ReLU) neural networks are widely used in machine learning.
  • Understanding the representational capacity of neural networks is crucial for their application.

Purpose of the Study:

  • To establish a theoretical connection between piecewise linear functions on polytope meshes and two-hidden-layer ReLU neural networks.
  • To provide precise bounds on the number of neurons required for such representations.
  • To explore the implications for analyzing the approximation capabilities of ReLU networks.

Main Methods:

  • Weak representation of continuous and discontinuous piecewise linear functions.
  • Analysis of functions defined on convex polytope meshes.
  • Derivation of neuron counts based on polytope and hyperplane counts.

Main Results:

  • Demonstrated that piecewise linear functions on convex polytope meshes can be weakly represented by two-hidden-layer ReLU networks.
  • Provided exact formulas for the number of neurons in each hidden layer.
  • Extended results to constant and linear finite element functions.
  • Discussed strict representation for tensor finite element functions using tensor neural networks.

Conclusions:

  • A bridge is established between ReLU neural networks and finite element functions.
  • This connection offers a new perspective for analyzing the approximation power of ReLU networks in L^p norms.
  • The findings are relevant for both theoretical computer science and applied mathematics.