folivo/PINA
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Update README.md

Browse Source
This commit is contained in:
Nicola Demo
2022-02-17 12:29:01 +01:00
committed by GitHub
parent b0c4f56d95
commit add909743b
1 changed files with 6 additions and 6 deletions
Download Patch File Download Diff File Expand all files Collapse all files

10
README.md
View File

@@ -37,8 +37,6 @@
<!-- * [Documentation](#documentation) -->
<!-- * [Testing](#testing) -->
* [Examples and Tutorials](#examples-and-tutorials)
<!-- * [Awards](#awards) -->
* [How to cite](#how-to-cite)
* [References](#references)
<!-- * [Recent works with PyDMD](#recent-works-with-pydmd) -->
* [Authors and contributors](#authors-and-contributors)
@@ -56,9 +54,10 @@ PINN is a novel approach that involves neural networks to solve supervised learn
First step is formalization of the problem in the PINA framework. We take as example here a simple Poisson problem, but PINA is already able to deal with **multi-dimensional**, **parametric**, **time-dependent** problems.
Consider:
<p align="center">
<img alt="Poisson approximation" src="readme/poisson_problem.png" width="80%" />
<img alt="Poisson approximation" src="readme/poisson_problem.png" width="50%" />
</p>
where *D* is a square domain, *Gamma*s are the boundaries and *u* the unknown field. The translation in PINA code becomes a new class containing all the information about the domain, about the `conditions` and nothing more:
```python
class Poisson(SpatialProblem):
spatial_variables = ['x', 'y']
@@ -85,6 +84,7 @@ class Poisson(SpatialProblem):
#### Problem solution
After defining it, we want of course to solve such a problem. The only things we need is a `model`, in this case a feed forward network, and some samples of the domain and boundaries, here using a Cartesian grid. In these points we are going to evaluate the residuals, which is nothing but the loss of the network.
```python
poisson_problem = Poisson()
@@ -100,9 +100,9 @@ pinn.train(1000, 100)
plotter = Plotter()
plotter.plot(pinn)
```
After the training we can infer our model, save it or just plot the PINN approximation.
After the training we can infer our model, save it or just plot the PINN approximation. Below the graphical representation of the PINN approximation, the analytical solution of the problem and the absolute error, from left to right.
<p align="center">
<img alt="Poisson approximation" src="readme/poisson_plot.png" width="80%" />
<img alt="Poisson approximation" src="readme/poisson_plot.png" width="100%" />
</p>