README.md

This repository was created to showcase an independent research report on the concept of Physics Informed Neural Networks (PINNs). Here I explain briefly the concepts of Backward Propagation, Forward Propagation, Loss Function formulation, optimization etc.

I then demonstrate a PiNNs approach to approximating the solutions to the 1D heat equation using PyTorch and compare the behavior of solutions based on the number of training steps.

Physics-Informed Neural Networks for 1D Heat Equation

We aim to solve the 1D heat equation using Physics-Informed Neural Networks. The mathematical model we are considering is:

The 1D heat equation is given by:

u_t = u_{xx}

Where:

• u_t denotes the partial derivative of u with respect to t.
• u_{xx} denotes the second partial derivative of u with respect to x.

Initial and Boundary Conditions

The initial condition is:

u(x, 0) = sin(πx)

And the boundary conditions are:

• At x = 0: u(0, t) = 0
• At x = 1: u(1, t) = 0

2D PDE Approximation Using PINNs: Navier-Stokes Equations

Consider the Navier-Stokes equations, which describe the motion of a fluid:

Navier-Stokes Equations:

• Momentum equation: ρ(∂u/∂t + u · ∇u) = -∇p + μ∇²u + f
• Continuity equation: ∇ · u = 0

Where:

• u is the velocity field.
• p is the pressure.
• ρ is the density.
• μ is the dynamic viscosity.
• f is the body force.

We implement the Physics-Informed Neural Network (PINN) solution to the 2D Navier-Stokes problem. You can find the code and plot the pressure field here.

To read further on my independent research on PINNs, please refer to this report.