heat-pinn

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
• Results

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:

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

Heat-PINN CLI

Installation

Build

.-----------------------------------------------------------------------.
|.##.....#.#######....###...#######........########.###.##....#.##....##|
|.##.....#.##........##.##.....##..........##.....#..##.###...#.###...##|
|.##.....#.##.......##...##....##..........##.....#..##.####..#.####..##|
|.########.######..##.....#....##...######.########..##.##.##.#.##.##.##|
|.##.....#.##......########....##..........##........##.##..###.##..####|
|.##.....#.##......##.....#....##..........##........##.##...##.##...###|
|.##.....#.#######.##.....#....##..........##.......###.##....#.##....##|
'-----------------------------------------------------------------------'

usage: heat build [-h] [--in-shape IN_SHAPE] [--out-shape OUT_SHAPE]
                  [--n-hidden-layers N_HIDDEN_LAYERS]
                  [--neuron-per-layer NEURON_PER_LAYER] [--actfun ACTFUN]
                  [--name NAME]

options:
  -h, --help            show this help message and exit
  --in-shape IN_SHAPE   Shape of the input tensor to feed into NN. Equal to
                        the number of independent variables of PDE.
  --out-shape OUT_SHAPE
                        Shape of the output tensor of NN. For heat transfer
                        it's T (equal to 1)
  --n-hidden-layers N_HIDDEN_LAYERS
                        Number of hidden layers in the NN
  --neuron-per-layer NEURON_PER_LAYER
                        Number of neurons in each hidden layer
  --actfun ACTFUN       Activation function
  --name NAME           Name of the model.

Train

.-----------------------------------------------------------------------.
|.##.....#.#######....###...#######........########.###.##....#.##....##|
|.##.....#.##........##.##.....##..........##.....#..##.###...#.###...##|
|.##.....#.##.......##...##....##..........##.....#..##.####..#.####..##|
|.########.######..##.....#....##...######.########..##.##.##.#.##.##.##|
|.##.....#.##......########....##..........##........##.##..###.##..####|
|.##.....#.##......##.....#....##..........##........##.##...##.##...###|
|.##.....#.#######.##.....#....##..........##.......###.##....#.##....##|
'-----------------------------------------------------------------------'

usage: heat train [-h] [--domain DOMAIN] [--boundary BOUNDARY] [--model MODEL]
                  [-l LR] [--epochs EPOCHS] [--every EVERY]

options:
  -h, --help            show this help message and exit
  --domain DOMAIN       Path of domain data file
  --boundary BOUNDARY   Path of boundary data file
  --model MODEL         Path of model file
  -l LR, --lr LR, --learning-rate LR
                        Learning rate for the optimizer
  --epochs EPOCHS       Number of training epochs
  --every EVERY         Print result for every n epochs

Inference

.-----------------------------------------------------------------------.
|.##.....#.#######....###...#######........########.###.##....#.##....##|
|.##.....#.##........##.##.....##..........##.....#..##.###...#.###...##|
|.##.....#.##.......##...##....##..........##.....#..##.####..#.####..##|
|.########.######..##.....#....##...######.########..##.##.##.#.##.##.##|
|.##.....#.##......########....##..........##........##.##..###.##..####|
|.##.....#.##......##.....#....##..........##........##.##...##.##...###|
|.##.....#.#######.##.....#....##..........##.......###.##....#.##....##|
'-----------------------------------------------------------------------'

usage: heat infer [-h] [--data DATA] [--model MODEL] [--output OUTPUT]

options:
  -h, --help       show this help message and exit
  --data DATA      Path of data file to perform inference
  --model MODEL    Path of model file
  --output OUTPUT  Path of output data file

Validation

Square geometry

Temperature profiles:

Results

Doughnut geometry

Screw

Connecting Rod

Gear 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.