A Physics-Informed Neural Network, to solve 2D steady-state heat equation based on the methodology introduced in: Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

• Introduction
• Results

In this project, a PINN is trained to solve a 2D heat equation and the final results is compared to a solution based on FDM method. For more detailts about the project read this.

The governing equation:

$$ \Theta = \frac{T - T_{\textbf{min}}}{T_{\textbf{max}}-T_{\textbf{min}}} $$

$$ \nabla^2{\Theta} = (\partial_{xx}+\partial_{yy})\Theta=0 $$

in the following domain:

$$ D = \{ (x, y)|-1\le x \le +1 \land -1\le y \le +1 \} $$

With the following boundary conditions:

$$ \begin{equation} \begin{cases} T(-1, y) = 75.0 \degree{C}\\ T(+1, y) = 0.0 \degree{C}\\ T(x, -1) = 50.0 \degree{C}\\ T(x, +1) = 0.0 \degree{C}\\ \end{cases} \end{equation} $$

When normalized:

$$ \begin{equation} \begin{cases} \Theta(-1, y) = 1\\ \Theta(+1, y) = 0\\ \Theta(x, -1) = \frac{2}{3}\\ \Theta(x, +1) = 0\\ \end{cases} \end{equation} $$

Results obtained from a 9 layered DNN (1000 epochs) and FDM code on a 100×100 grid. The FDM code is written in Python.

This implementation is based on Tensorflow 2.0 package and made possible by Google Colabratory GPU.