Physics Informed Neural Network (PINN)

Mass-spring-damper PDE

Exploring simple implementation of PINNs into a damped spring-mass system using pytorch as framework.

This code was largely based on Ben Moseley's workshop benmoseley. He also launched a video in youtube explaining youtube.

Here I'll explore a bit more this system including a different range of forced inputs as well as exploring the hyperparameters of the system with a simple grid search.

In this project the main objective is to discover the parameters of the equation described bellow (b and k, assuming m = 1 Kg).

Description

Given a Spring-mass system following the equation:

$$m\frac{\partial^2 x(t)}{\partial t^2} + b\frac{\partial x(t)}{\partial t} + kx(t) = 0$$

If we assume m = 1 Kg and b and k as the following: $$m = 1 $$ $$b = 2\xi\omega_0 (b=\mu \ in \ the \ code)$$ $$k = w_0^2$$

We get the homogeneous equation bellow: $$\frac{\partial^2 x(t)}{\partial t^2} + 2\xi\omega_0\frac{\partial x(t)}{\partial t} + w_0^2x(t) = 0$$

Litte gif showing the form of the position $x(t)$ of the spring and its derivative $\frac{\partial x(t)}{\partial t}$ over time for the homogeneous equation above:

Now let's assume we want to approximate the underlying solution $x(t)$ of the equation as a Fully Connected Neural Network where its enteries are time points

$$t = \begin{pmatrix}t_0\\\vdots\\t_k\end{pmatrix}$$

and its outputs are the values of $x(t)$

$$x(t) = \begin{pmatrix}x(t_0)\\\vdots\\x(t_k)\end{pmatrix}$$

for any given t in the domain $t \in D = [a,b]$. So $NN(t) \approx x(t), \forall t \in D$.

Also let's add the parameters we want to discover as "weights" in the optimization of the Neural Network. So b and k are also added as learnable parameters in the training.

self.k_guess = torch.nn.Parameter(torch.tensor([float(pinn_params["k_guess"])], requires_grad=True))
self.mu_guess = torch.nn.Parameter(torch.tensor([float(pinn_params["mu_guess"])], requires_grad=True))

self.optimiser = torch.optim.Adam(list(self.pinn.parameters())+[self.k_guess, self.mu_guess],lr=self.learning_rate, betas=(0.95, 0.999))

Examples

References

[1] Moseley, B (2022). Physics-informed machine learning: from concepts to real-world applications, University of Oxford

[2] Maziar Raissi, Paris Perdikaris, and George Em Karniadakis (2017). Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

Authors

• @jeduapf

Tags