The repository contains the code and results for the PyTorch Implementation of the paper titled Artificial Neural Networks for Solving Ordinary and Partial Differential Equations. Further, Euler Method, a popular numerical method for solving differential equations has been implemented as well.
- First Order ODE
- Second Order ODE
- PDE with Dirichlet Boundary Conditions
- PDE with Mixed Boundary Conditions
- Equation:
$\frac{dy}{dx}+y=\exp(\frac{-x}{5})\cos(x)$ - Initial Condition:
$y(0)=0$ - Neural Trial Solution:
$y=A+xN(x,\theta)$ ,$N(x,\theta)$ is the Neural Function with parameter$\theta$
- Equation:
$\frac{d^2y}{dx^2}+\frac{1}{5}\frac{dy}{dx}+y=\exp(\frac{-x}{5})\cos(x)$ - Initial Conditions:
$y(0)=0,y'(0)=1$ - Neural Trial Solution:
$y=A+A_1x+x^2N(x,\theta)$
- Equation:
$\nabla^2\psi(x,y)=\exp(-x)(x-2+y^3+6y)$ - Boundary Conditions:
$\psi(0,y)=y^3,\psi(1,y)=\exp(-1)(1+y^3),\psi(x,0)=x\exp(-x)$ $,\psi=\exp(-x)(x+1)$ - Neural Trial Solution:
$\psi=A(x,y)+x(1-x)y(1-y)N(x,y,\theta)$
