Approximation Theory for Neural Networks: Old and New

Dimensions overall score 3.0

• Surveys the approximation theory of neural networks, covering classical density results, quantitative bounds, depth-width trade-offs, and recent developments in Kolmogorov-Arnold Networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces.problem 35%
  Evidencepartial

  Surveys the approximation theory of neural networks, covering classical density results, quantitative bounds, depth-width trade-offs, and recent developments in Kolmogorov-Arnold Networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces.

  Implicationpartial

  Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.

  Verificationpartial

  partial

• Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces.method 35%
  Evidencepartial

  Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces.

  Implicationpartial

  Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.

  Verificationpartial

  partial

• ScienceToStartup currently rates this 3.0/10 on the public viability pass. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. Code availability is flagged in the production record; the public repository link still needs proof alignment.result 35%
  Evidencepartial

  ScienceToStartup currently rates this 3.0/10 on the public viability pass. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. Code availability is flagged in the production record; the public repository link still needs proof alignment.

  Implicationpartial

  Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.

  Verificationpartial

  partial

• Neural Network Theory moved forward this cycle; last verified May 2026. Public score 3.0/10. Production flags indicate code availability.why now 35%
  Evidencepartial

  Neural Network Theory moved forward this cycle; last verified May 2026. Public score 3.0/10. Production flags indicate code availability.

  Implicationpartial

  Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.

  Verificationpartial

  partial

Competitive landscape

Surveys the approximation theory of neural networks, covering classical density results, quantitative bounds, depth-width trade-offs, and recent developments in Kolmogorov-Arnold Networks.