PyTorch Implementation to solve Differential Equations using Neural networks.

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.

Implemented

• First Order ODE
• Second Order ODE
• PDE with Dirichlet Boundary Conditions
• PDE with Mixed Boundary Conditions

Results

First Order ODE

Equation Considered and 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 $\theta$

Graph of Neural and Analytic Solution

Second Order ODE

Equation Considered and Conditions

• 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: $y=A+A_1x+x^2N(x,\theta)$

Graph of Neural and Analytic Solution

Partial Differential Equations

Equation Considered and Dirichlet Boundary Conditions

• 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: $\psi=A(x,y)+x(1-x)y(1-y)N(x,y,\theta)$

Graph of Neural and Analytic Solution