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Stokes Equation Plan

Β· 2 min read
Brahi
Brahi
author

On January 1, 2025, I set out to write an article about returning to mathematics. This article is a collection of all the topics I've learned, both in my undergraduate degree in applied mathematics and my master's degree in software engineering, although I believe my free time has been my greatest source of learning.

I made a list for developing experiments that can approximate Stokes' equation using neural networks.

This list, like this post, may change, but it will always retain its essence

Click on Read More to see the list.

Functional Analysis​

  • 🟑 Normed spaces and preliminaries
    • 🟑 Comment on having a solid foundation in metric spaces, vector spaces, and continuity definitions for this course
    • πŸ”΅ Lebesgue space -🟑 Normed spaces
      • πŸ”΅ Definition
      • 🧩 Sequentice and dense
      • 🧩 Continuous function
      • πŸ§©πŸ”„ Complete sets and Compact sets?
  • β›” Banach and Hilbert Space
    • 🧩 Banach spaces
    • 🧩 Hilbert spaces
    • 🧩 Examples
    • β›” Dual?
    • β›” the Riesz Representation Theorem
    • β›” Hahn-Banach Extension Theorem
  • β›” Distributions and Sobolev Spaces
    • β›” Definition
    • β›” Test functions?
    • β›” Distributions
    • β›” Locally integrable functions?
    • β›” Derivatives of distributions
    • β›” Weak derivatives
    • β›” Sobolev Norms
    • β›” Sobolev Spaces
    • β›” Poincare inequality
  • β›” Weak solution of the Poisson equation
    • β›” formulationn equation
    • β›” Dirichlet integral develop
    • β›” Weak solution of the Laplace equation
    • β›” Weak solution of the Poisson equation

Universal Approximation Theorem​

  • βœ… Neural networks
  • 🟑Theorems
    • 🟑 Spaces pf continuos functions
    • 🟑 Lp(Ξ©)L^p(\Omega) spaces
    • β›” Sobolev spaces
    • 🧩 Multilayer networks

Computational Experiments​

  • 🧩 Neural network implementation
    • 🧩 f:Rβ†’Rf:\mathbb{R}\to \mathbb{R}
      • 🧩 continuous function xsin⁑(x)x\sin(x)
      • 🧩 Step function not Lebesgue integrable on [0,1][0,1] but if on [0,c][0,c] when c<1c<1
      • β›” Sobolev function
    • β›” f:R2β†’Rf:\mathbb{R}^2\to \mathbb{R}
    • β›” f:R3β†’Rf:\mathbb{R}^3\to \mathbb{R}
  • β›” Mesh
  • β›” Approximating Differential Equations with Neural Networks
    • β›” EDO: harmonic oscillator?
    • β›” EDO: forced harmonic oscillator?
    • β›” EDP: Heat equation?
    • β›” EDP: Laplace equation
    • β›” EDP: Poisson equation

Stokes Equations​

  • β›” Formalization
  • β›” Variational formulation
  • β›” Theorems
  • β›” Implementation
  • β›” Examples