🍔 $\textbf{Burger-PINN}$ ⚛️

A Physics-Informed Neural Network to solve Burgers' equation.

Reference: Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
Author's Github page: https://maziarraissi.github.io/PINNs/

Check out my other PINN project: heat-pinn

Introduction

This work is an implementation of the reference paper above. In this work, a neural network is trained to solve Burgers' equation, a well-known PDE, for a given set of BCs and an IC.

To train this model, certain number of points on the both boundaries and on the initial timestep were sampled, as input data, called data points, to compute MSE loss function, in addition to another set of points on the main domain of the problem, or collocation points to penalize the network in case the prediction on this set is not consistent with the governing PDE.

Problem details

The governing PDE is:

$\partial_{t}{u}+u\partial_{x}{u}-(\frac{0.01}{\pi})\partial_{xx}{u} = 0$

Which is solved in the following spatial and temporal domain:

$D = \{ (t, x)\ |\ 0\le t \le +1 \land -1\le x \le +1 \}$

Regarding the following initial and boundary conditions:

$$ \begin{equation} \begin{cases} u(0, x) = -\sin({\pi x})\\ u(t, +1) = 0.0 \\ u(t, -1) = 0.0 \\ \end{cases} \end{equation} $$

Results

$u(t, x)$ profile (Figure 1. in reference or author's Github page)

$u(t, x)$ in Cartesian coordinates for $t \in \{0.25,0.5,0.75\}$ (Figure 1. in reference or author's Github page)

$u(t, x)$ in Cartesian coordinates for more t values

Note

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