A Physics-Informed Neural Network to solve 2D steady-state heat equation.
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
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:
$$
\begin{equation}
\begin{cases}
\Theta(-1, y) = 1\
\Theta(+1, y) = 0\
\Theta(x, -1) = \frac{2}{3}\
\Theta(x, +1) = 0\
\end{cases}
\end{equation}
$$
Temperature profiles:
Results obtained from a 9 layered DNN (1000 epochs) and FDM code on a 100×100 grid. The FDM code is written in Python.
Method | Computation time (s) |
---|---|
PINN | 66.35 |
FDM | 77.60 |
This implementation is based on Tensorflow 2.0 package and made possible by Google Colabratory GPU.