Neural Networks for Scientific Computing

Introduction to Neural Networks

Neural networks are powerful tools for solving complex problems in scientific computing and physics.

import tensorflow as tf
import numpy as np

# Create a simple neural network
def create_model():
    model = tf.keras.Sequential([
        tf.keras.layers.Dense(64, activation='relu', input_shape=(1,)),
        tf.keras.layers.Dense(32, activation='relu'),
        tf.keras.layers.Dense(1)
    ])
    
    model.compile(optimizer='adam', loss='mse')
    return model

# Generate training data
x = np.linspace(-5, 5, 1000)
y = x**2 + np.random.normal(0, 0.1, 1000)

# Train model
model = create_model()
model.fit(x, y, epochs=100, validation_split=0.2)
                    

Advanced Architectures

Explore different neural network architectures for scientific applications.

import tensorflow as tf

# Create a CNN for image analysis
def create_cnn_model():
    model = tf.keras.Sequential([
        tf.keras.layers.Conv2D(32, (3, 3), activation='relu', 
                             input_shape=(64, 64, 1)),
        tf.keras.layers.MaxPooling2D((2, 2)),
        tf.keras.layers.Conv2D(64, (3, 3), activation='relu'),
        tf.keras.layers.MaxPooling2D((2, 2)),
        tf.keras.layers.Flatten(),
        tf.keras.layers.Dense(64, activation='relu'),
        tf.keras.layers.Dense(1)
    ])
    
    model.compile(optimizer='adam', loss='mse')
    return model

# Generate 2D physics data (e.g., field distributions)
def generate_field_data(n_samples=1000):
    x = np.linspace(-5, 5, 64)
    y = np.linspace(-5, 5, 64)
    X, Y = np.meshgrid(x, y)
    
    # Example: Electric field from point charge
    R = np.sqrt(X**2 + Y**2)
    field = 1 / (R + 1e-6)
    
    return field.reshape(64, 64, 1)
                    

Custom Loss Functions

Learn how to incorporate physical constraints into neural network training.

import tensorflow as tf

class PhysicsModel(tf.keras.Model):
    def __init__(self):
        super(PhysicsModel, self).__init__()
        self.dense1 = tf.keras.layers.Dense(64, activation='relu')
        self.dense2 = tf.keras.layers.Dense(32, activation='relu')
        self.dense3 = tf.keras.layers.Dense(1)
        
    def call(self, inputs):
        x = self.dense1(inputs)
        x = self.dense2(x)
        return self.dense3(x)
    
    def physics_loss(self, y_true, y_pred):
        # Example: Conservation of energy constraint
        dy_dx = tf.gradients(y_pred, self.inputs)[0]
        physics_constraint = tf.reduce_mean(tf.square(dy_dx + y_pred))
        mse = tf.reduce_mean(tf.square(y_true - y_pred))
        return mse + 0.1 * physics_constraint

# Create and train model
model = PhysicsModel()
model.compile(optimizer='adam', loss=model.physics_loss)
                    

Practice Exercises

Additional Resources

• TensorFlow Custom Training
• PyTorch Tutorials
• Neural Networks in Physics Research