A Physics-Informed Neural Network to solve 2D steady-state heat equations.

Project README

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 100Ã—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.

Open Source Agenda is not affiliated with "Heat Pinn" Project. README Source: 314arhaam/heat-pinn

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