Heat Pinn Save

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

Project README

🔥 $\textbf{Heat-PINN}$ 🔥

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

Table of Contents

Introduction

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.

Problem

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

Square geometry

Temperature profiles:

Doughnut geometry

Performance Comparison

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

Note

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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