Solving Differential Equations with Neural Networks

Reproduction of the papers:

[1] Lagaris, I., Likas, A., & Fotiadis, D. 1997, Computer Physics Communications,104, 1

Roadmap:

The repository contains three parts:

• Tensorflow version - code based on the Tensorflow library, enables simple :

  • Works:
    • ODEs with easy Dirichlet and Neumann initial conditions (Examples no. 1-3),
    • Systems of coupled ODEs (Example no. 4),
    • PDEs with Dirichlet boundary condition (Example no. 5),
  • To be done:
    • Neumann boundary conditions for PDEs (Examples no. 6-7 - written, not converging),
    • Custom weights initialization,
    • Better optimization procedure (BFGS),

• Numpy_version - code from scratch based only on numpy for solving differential equations via the trial solution with a shallow network:

  • Works:
    • ODEs with Dirichlet and Neumann boundary conditions Examples (no. 1-3),
    • PDEs with Dirichlet Conditions (Example no. 5),
  • To be done:
    • Neumann boundary conditions for PDEs,
    • Systems of coupled ODEs,
    • optimization of the tensor operations,

• Others - some other experiments:

  • Keras version - very inelastic and inelegant code.