Step-by-Step Tutorial on Physics Informed Neural Networks (PINNs) for Ordinary Differential Equations

Introduction:

Physics Informed Neural Networks (PINNs) are a class of neural networks that are designed to solve partial differential equations (PDEs) or ordinary differential equations (ODEs) by incorporating them into the loss function of the neural network. This allows the neural network to learn the underlying physics of the problem and make predictions that are consistent with the physics.

In this tutorial, we will focus on using PINNs to solve ordinary differential equations (ODEs). We will walk through the process step-by-step, from defining the ODE problem to training the neural network and evaluating its performance.

Step 1: Define the Ordinary Differential Equation

For this tutorial, let’s consider a simple first-order ODE:

dy/dx = x^2

We want to find the solution y(x) that satisfies this ODE for x in the range [0, 1].

Step 2: Define the Loss Function

The loss function for PINNs consists of two components: the data loss and the physics loss. The data loss enforces the neural network to fit the data (initial and boundary conditions), while the physics loss enforces the neural network to satisfy the ODE.

For this example, let’s consider the initial condition y(0) = 1 and the boundary condition y(1) = 1.

The total loss function can be defined as:

L = L_data + L_physics

where the data loss is given by:

L_data = (y_pred(0) – 1)^2 + (y_pred(1) – 1)^2

and the physics loss is given by:

L_physics = (dy/dx – x^2)^2

Step 3: Define the Neural Network Architecture

Next, we need to define the neural network architecture that will be used to approximate the solution to the ODE. For simplicity, let’s consider a feedforward neural network with one hidden layer and a tanh activation function.

Step 4: Train the Neural Network

Now, we can train the neural network using stochastic gradient descent or any other optimization algorithm. During training, we minimize the total loss function defined in Step 2.

Step 5: Evaluate the Performance

Once the neural network is trained, we can evaluate its performance by comparing its predictions with the true solution to the ODE. We can calculate the mean squared error or other metrics to assess the accuracy of the predictions.

Conclusion:

In this tutorial, we have walked through the steps of solving an ordinary differential equation using Physics Informed Neural Networks (PINNs). By incorporating the ODE into the loss function of the neural network, we were able to learn the underlying physics of the problem and make accurate predictions. PINNs offer a powerful and versatile approach to solving differential equations and can be applied to a wide range of physical and engineering problems.