Physics-informed Neural Networks for Fluid Dynamics

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Now, if you’re like me and have a love/hate relationship with math and physics, this might sound intimidating at first. But trust me, it’s not as scary as it seems! In fact, PINNs are actually pretty simple to understand once you break them down into their basic components.

So what exactly is a PINN? Well, let’s start with the basics neural networks (NNs) are essentially just fancy algorithms that can learn patterns and make predictions based on data. They work by taking input data (like images or text), feeding it through a series of mathematical functions called layers, and then outputting an answer based on what they’ve learned from previous examples.

But here’s the cool part PINNs take things to the next level by incorporating physics into the mix! That means instead of just learning patterns like regular NNs do, PINNs can actually solve complex physical problems using their neural network magic.

For example, let’s say you want to simulate fluid flow around an object (like a car or airplane wing). With traditional methods, this would involve solving a bunch of complicated equations and running simulations for hours on end. But with PINNs, all you need is some input data (like the shape of the object) and a neural network that can learn how to solve those equations!

Here’s an example code snippet using Python:

# Import necessary libraries
import numpy as np
from scipy import integrate
import matplotlib.pyplot as plt
from sklearn.metrics import mean_squared_error

# Define the physical problem (in this case, a simple heat equation)
def f(x, t):
    return -k * u(x, t) + q(x, t)

# Define the input data (like the shape of an object or initial conditions)
x = np.linspace(-10, 10, num=500) # Create an array of 500 evenly spaced numbers between -10 and 10
t = np.linspace(0, 2*np.pi, num=100) # Create an array of 100 evenly spaced numbers between 0 and 2*pi
u_init = np.sin(x) * np.cos(t) # Calculate the initial conditions using sine and cosine functions
q = lambda x, t: -k * u_init(x, t) + f(x, t) # Define the input data using the initial conditions and the physical problem function

# Define the neural network (using Keras and TensorFlow)
from tensorflow import keras
import tensorflow as tf

model = keras.Sequential([
    # Input layer with 2 neurons for x and t coordinates
    keras.Input(shape=(2,)),
    # Hidden layers with 10 neurons each (you can adjust this based on the complexity of your problem)
    keras.layers.Dense(units=10),
    keras.layers.Activation('relu'),
    keras.layers.Dense(units=10),
    keras.layers.Activation('relu'),
    # Output layer with 1 neuron for the solution value at each point in space and time
    keras.layers.Dense(units=1)
])

# Define the loss function (using mean squared error between predicted and actual solutions)
def loss_function(y_true, y_pred):
    return tf.keras.losses.mean_squared_error(y_true, y_pred)

# Train the neural network using backpropagation to minimize the loss function (using Adam optimizer and 10 epochs)
model.compile(optimizer='adam', loss=loss_function)
history = model.fit(x=np.stack([x, t], axis=-1), y=u_init, batch_size=32, epochs=10) # Train the model using the input data and the defined loss function for 10 epochs

# Evaluate the neural network on a test dataset (using mean squared error between predicted and actual solutions)
test_data = np.stack([np.linspace(-10, 10, num=50), np.linspace(0, 2*np.pi, num=100)], axis=-1) # Create a test dataset using the same range of values as the input data
predictions = model.predict(x=test_data) # Use the trained model to make predictions on the test dataset
mse = mean_squared_error(y_true=u_init[..., None].reshape(-1), y_pred=predictions.reshape(-1)) # Calculate the mean squared error between the predicted and actual solutions
print("Mean squared error:", mse) # Print the mean squared error as a measure of the model's accuracy

And that’s it! With just a few lines of code, you can simulate fluid flow around an object using PINNs and neural networks. Pretty cool, huh?

Of course, there are still some limitations to this approach for example, PINNs may not be as accurate or efficient as traditional methods in certain cases (especially when dealing with complex physical problems). But overall, they offer a powerful new tool for solving real-world engineering and scientific challenges!

It’s not as scary as it sounds, I promise! Give it a try and let us know what you think in the comments below.

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