PINN for biological systems

This repository tries to solve systems of differential equation for biological systems using PINNs (Physics Informed Neural Networks).
The scripts are implemented using Pytorch or the ad hoc library DeepXDE.

Organization of the repository

Up to now therepository contain three main files: first_pinn.py, pinn_deepxde.py and inverse_pinn.py.
The first two scripts solve the exact same problem with same parameters, the only difference is that in first_pinn.py I solved the ploblem 'from scratch' using only Pytorch, while in pinn_deepxde.py the problem is solved using the DeepXDE library.
The third file (inverse_pinn.py) inplements the inverse pinn problem where the aim is to infer some parameters not known using data.

Problem

The differential equation system that I solved in the two scripts is the following:

$$\begin{cases} \frac{d}{dt}x_1 = 0\\\ \frac{d}{dt}x_2 = \lambda x_1 + (\lambda - \nu) x_2\\\ \frac{d}{dt}y_1 = \nu x_2 - \gamma y_1\\\ \frac{d}{dt}z = 2\gamma y_1 - \delta z \end{cases}$$

This system of equation is a semplified version of the evolution of stem cells. For further details you can refer to this paper. The values of the parameters that where used are:

• $\lambda = 0.2$
• $\nu = 0.33$
• $\gamma = 2.0$
• $\delta = 0.33$

With the following initial conditions:

• $x_1 = 6$
• $x_2 = 5$
• $y_1 = 0$
• $z = 0$

Inverse problem

In the inverse problem, so in the inverse_pinn.py file, $\nu$ and $\delta$ were assumed already known and $\lambda$ and $\gamma$ unknown. The aim was to infer this two parameter using noisy data generated with scipy.integrate.odeint, the noise follows a Gaussian distribution with $\mu = 0$ and $\sigma = 0.5$.
The generated data used by the neural network for the training include only the solution for $y_1$ and $z$. Nonetheless the parameters were succesfully inferred.

How to use

To run the simulation you have to clone this repository in an empty folder: from the terminal (inside an empty folder) you can use the following command

git clone https://github.com/TommyGiak/biological_PINN.git

then you can just run with python the script:

python first_pinn.py

or

python pinn_deepxde.py

for the direct problem,

python inverse_pinn.py

for the inverse one.