Understanding Neural Networks: From Neurons to Transformers

Introduction

This guide builds your understanding of neural networks from the ground up, starting with how individual neurons process data, scaling to vectorized layer operations, and culminating in transformer architectures with attention mechanisms.

flowchart LR
    start["🎯 Start Here"] --> part1["Part 1<br/>Single Neuron<br/>────<br/>Activation<br/>Functions"]
    part1 --> part2["Part 2<br/>Layers<br/>────<br/>Vectorization<br/>& Depth"]
    part2 --> part3["Part 3<br/>Attention<br/>────<br/>Context-Aware<br/>Processing"]
    part3 --> part4["Part 4<br/>Transformers<br/>────<br/>Complete<br/>Architecture"]
    part4 --> end_goal["🎓 Goal Achieved<br/>────<br/>Understand GPT,<br/>BERT, GFMs"]
    
    style start fill:#e1f5ff
    style part1 fill:#e8f0ff
    style part2 fill:#f0e1ff
    style part3 fill:#ffe8e1
    style part4 fill:#ffe1e1
    style end_goal fill:#e1ffe1

Learning Roadmap: Our Journey Through Neural Networks

What you’ll learn: - How neurons transform data with activation functions - How vectorization makes processing efficient - Why attention is revolutionary for sequence modeling - How transformers combine these ideas into powerful models

Video Resources

For excellent visual explanations of neural networks, check out 3Blue1Brown’s Neural Networks series:

• But what is a neural network? - Foundation concepts
• Gradient descent, how neural networks learn - Training process

---

Part 1: The Individual Neuron

What Does a Neuron Do?

At its core, a neuron is a simple computational unit that:

1. Takes multiple inputs
2. Combines them with learned weights
3. Adds a bias term
4. Passes the result through an activation function

flowchart LR
    subgraph inputs["Inputs"]
        x1["x₁ = 1.5"]
        x2["x₂ = 2.0"]
        x3["x₃ = -0.5"]
    end
    
    subgraph weights["× Weights"]
        w1["w₁ = 0.5"]
        w2["w₂ = -1.0"]
        w3["w₃ = 0.3"]
    end
    
    subgraph computation["Weighted Sum"]
        sum["Σ(xᵢ × wᵢ) + b"]
        bias["+ bias (0.5)"]
    end
    
    subgraph activation["Activation"]
        z["z = -0.35"]
        act["f(z)"]
        output["Output"]
    end
    
    x1 --> w1
    x2 --> w2
    x3 --> w3
    w1 --> sum
    w2 --> sum
    w3 --> sum
    bias --> sum
    sum --> z
    z --> act
    act --> output
    
    style inputs fill:#e1f5ff
    style weights fill:#fff3e1
    style computation fill:#f0e1ff
    style activation fill:#e1ffe1

Single Neuron Computation Flow

Key insight: A neuron is just a weighted sum followed by a non-linear function. That’s it!

Let’s see this in action with code:

import numpy as np
import matplotlib.pyplot as plt

# A single neuron processes inputs
def single_neuron(inputs, weights, bias, activation_fn):
    """
    Simulate a single neuron's computation.
    
    Args:
        inputs: array of input values
        weights: array of weights (same length as inputs)
        bias: single bias value
        activation_fn: function to apply to weighted sum
    
    Returns:
        The neuron's output after activation
    """
    # Step 1: Weighted sum
    z = np.dot(inputs, weights) + bias
    
    # Step 2: Apply activation function
    output = activation_fn(z)
    
    return output, z

# Example inputs and parameters
inputs = np.array([1.5, 2.0, -0.5])
weights = np.array([0.5, -1.0, 0.3])
bias = 0.5

# Without activation (just linear combination)
output_linear, z = single_neuron(inputs, weights, bias, lambda x: x)
print(f"Input values: {inputs}")
print(f"Weights: {weights}")
print(f"Bias: {bias}")
print(f"\nWeighted sum (z): {z:.3f}")
print(f"Linear output: {output_linear:.3f}")

Input values: [ 1.5  2.  -0.5]
Weights: [ 0.5 -1.   0.3]
Bias: 0.5

Weighted sum (z): -0.900
Linear output: -0.900

What to notice: The neuron computes a weighted sum of its inputs plus a bias. Without an activation function, this is just a linear transformation—not very powerful for learning complex patterns.

Activation Functions: Adding Non-Linearity

Activation functions introduce non-linearity, which is crucial for neural networks to learn complex patterns. Let’s explore the most common ones:

# Define common activation functions
def relu(x):
    """ReLU: Rectified Linear Unit"""
    return np.maximum(0, x)

def sigmoid(x):
    """Sigmoid: Squashes values to (0, 1)"""
    return 1 / (1 + np.exp(-x))

def tanh(x):
    """Tanh: Squashes values to (-1, 1)"""
    return np.tanh(x)

def leaky_relu(x, alpha=0.01):
    """Leaky ReLU: Like ReLU but allows small negative values"""
    return np.where(x > 0, x, alpha * x)

# Visualize these functions
x = np.linspace(-5, 5, 100)

fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# ReLU
axes[0, 0].plot(x, relu(x), 'b-', linewidth=2)
axes[0, 0].grid(True, alpha=0.3)
axes[0, 0].set_title('ReLU: max(0, x)', fontsize=12, fontweight='bold')
axes[0, 0].set_xlabel('Input (x)')
axes[0, 0].set_ylabel('Output')
axes[0, 0].axhline(y=0, color='k', linewidth=0.5)
axes[0, 0].axvline(x=0, color='k', linewidth=0.5)

# Sigmoid
axes[0, 1].plot(x, sigmoid(x), 'g-', linewidth=2)
axes[0, 1].grid(True, alpha=0.3)
axes[0, 1].set_title('Sigmoid: 1/(1+e^(-x))', fontsize=12, fontweight='bold')
axes[0, 1].set_xlabel('Input (x)')
axes[0, 1].set_ylabel('Output')
axes[0, 1].axhline(y=0.5, color='r', linewidth=0.5, linestyle='--', alpha=0.5)

# Tanh
axes[1, 0].plot(x, tanh(x), 'r-', linewidth=2)
axes[1, 0].grid(True, alpha=0.3)
axes[1, 0].set_title('Tanh: (e^x - e^(-x))/(e^x + e^(-x))', fontsize=12, fontweight='bold')
axes[1, 0].set_xlabel('Input (x)')
axes[1, 0].set_ylabel('Output')
axes[1, 0].axhline(y=0, color='k', linewidth=0.5)

# Leaky ReLU
axes[1, 1].plot(x, leaky_relu(x), 'm-', linewidth=2)
axes[1, 1].grid(True, alpha=0.3)
axes[1, 1].set_title('Leaky ReLU: max(0.01x, x)', fontsize=12, fontweight='bold')
axes[1, 1].set_xlabel('Input (x)')
axes[1, 1].set_ylabel('Output')
axes[1, 1].axhline(y=0, color='k', linewidth=0.5)
axes[1, 1].axvline(x=0, color='k', linewidth=0.5)

plt.tight_layout()
plt.show()

What to notice:

• ReLU is the most popular: simple, fast, and effective. It “turns off” negative values completely.
• Sigmoid squashes values between 0 and 1, useful for probabilities.
• Tanh centers outputs around 0, often better than sigmoid for hidden layers.
• Leaky ReLU prevents “dead neurons” by allowing small negative values.

Seeing the Effect of Activation Functions

Let’s see how different activation functions transform our neuron’s output:

# Use the same inputs from before
inputs = np.array([1.5, 2.0, -0.5])
weights = np.array([0.5, -1.0, 0.3])
bias = 0.5

# Compute weighted sum
z = np.dot(inputs, weights) + bias

print(f"Weighted sum (z): {z:.3f}\n")

# Apply different activations
activations = {
    'Linear (no activation)': lambda x: x,
    'ReLU': relu,
    'Sigmoid': sigmoid,
    'Tanh': tanh,
    'Leaky ReLU': leaky_relu
}

for name, fn in activations.items():
    output = fn(z)
    print(f"{name:25s}: {output:.3f}")

Weighted sum (z): -0.900

Linear (no activation)   : -0.900
ReLU                     : 0.000
Sigmoid                  : 0.289
Tanh                     : -0.716
Leaky ReLU               : -0.009

Why this matters: The activation function dramatically changes the neuron’s output. This non-linear transformation is what allows neural networks to learn complex, non-linear patterns in data.

---

Part 2: From Single Neurons to Layers

The Power of Vectorization

Rather than computing neurons one at a time, we can process an entire layer simultaneously using matrix operations. This is both computationally efficient and conceptually elegant.

flowchart TB
    subgraph single["Single Neuron (Sequential)"]
        direction LR
        i1["Input<br/>Vector<br/>(3 dims)"] --> n1["Neuron 1"]
        i2["Input<br/>Vector<br/>(3 dims)"] --> n2["Neuron 2"]
        i3["Input<br/>Vector<br/>(3 dims)"] --> n3["Neuron 3"]
        n1 --> o1["Output 1"]
        n2 --> o2["Output 2"]
        n3 --> o3["Output 3"]
    end
    
    subgraph vectorized["Vectorized Layer (Parallel)"]
        direction LR
        iv["Input<br/>Vector<br/>(3 dims)"] --> matrix["Weight Matrix<br/>(3 × 5)<br/>+ Bias<br/>+ Activation"]
        matrix --> ov["Output<br/>Vector<br/>(5 dims)"]
    end
    
    single -.->|"Matrix Operation"| vectorized
    
    style single fill:#ffe1e1
    style vectorized fill:#e1ffe1
    style matrix fill:#fff3e1

From Single Neuron to Layer of Neurons

Key insight: Instead of looping through neurons, one matrix multiplication processes all neurons simultaneously!

Video Resource

For understanding how layers work together, see 3Blue1Brown’s Neural network intuitions (timestamp 7:00 onwards).

# A layer is just multiple neurons working in parallel
class NeuralLayer:
    """A single layer of neurons with vectorized operations."""
    
    def __init__(self, n_inputs, n_neurons):
        """
        Initialize a layer.
        
        Args:
            n_inputs: number of input features
            n_neurons: number of neurons in this layer
        """
        # Each neuron has n_inputs weights
        # Shape: (n_inputs, n_neurons)
        self.weights = np.random.randn(n_inputs, n_neurons) * 0.1
        
        # Each neuron has one bias
        # Shape: (n_neurons,)
        self.biases = np.zeros(n_neurons)
    
    def forward(self, inputs, activation_fn=relu):
        """
        Forward pass through the layer.
        
        Args:
            inputs: input array of shape (n_samples, n_inputs)
            activation_fn: activation function to apply
        
        Returns:
            outputs after activation, shape (n_samples, n_neurons)
        """
        # Matrix multiplication: (n_samples, n_inputs) @ (n_inputs, n_neurons)
        # Result: (n_samples, n_neurons)
        z = np.dot(inputs, self.weights) + self.biases
        
        # Apply activation function element-wise
        return activation_fn(z)

# Create a layer with 3 inputs and 5 neurons
layer = NeuralLayer(n_inputs=3, n_neurons=5)

# Process a batch of 4 samples
batch_inputs = np.random.randn(4, 3)
outputs = layer.forward(batch_inputs, activation_fn=relu)

print(f"Input shape: {batch_inputs.shape}")
print(f"Weight matrix shape: {layer.weights.shape}")
print(f"Output shape: {outputs.shape}\n")

print("Sample input:")
print(batch_inputs[0])
print("\nCorresponding output:")
print(outputs[0])

Input shape: (4, 3)
Weight matrix shape: (3, 5)
Output shape: (4, 5)

Sample input:
[-0.09917586  1.846637   -1.07008477]

Corresponding output:
[0.         0.14226798 0.04979847 0.         0.07559149]

What to notice:

• The weight matrix has shape (n_inputs, n_neurons) - each column represents one neuron’s weights.
• One matrix multiplication processes all neurons and all samples simultaneously.
• Output shape is (n_samples, n_neurons) - each sample gets transformed into a vector of neuron activations.

Building a Multi-Layer Network

Now let’s stack multiple layers to create a deep neural network:

flowchart LR
    subgraph input["Input Layer"]
        i1["x₁"]
        i2["x₂"]
        i3["x₃"]
    end
    
    subgraph hidden1["Hidden Layer 1<br/>(8 neurons)"]
        h11["🔵"]
        h12["🔵"]
        h13["🔵"]
        h14["🔵"]
        h15["🔵"]
        h16["🔵"]
        h17["🔵"]
        h18["🔵"]
    end
    
    subgraph hidden2["Hidden Layer 2<br/>(5 neurons)"]
        h21["🟢"]
        h22["🟢"]
        h23["🟢"]
        h24["🟢"]
        h25["🟢"]
    end
    
    subgraph output["Output Layer<br/>(2 neurons)"]
        o1["🔴"]
        o2["🔴"]
    end
    
    i1 --> h11 & h12 & h13 & h14 & h15 & h16 & h17 & h18
    i2 --> h11 & h12 & h13 & h14 & h15 & h16 & h17 & h18
    i3 --> h11 & h12 & h13 & h14 & h15 & h16 & h17 & h18
    
    h11 & h12 & h13 & h14 & h15 & h16 & h17 & h18 --> h21 & h22 & h23 & h24 & h25
    
    h21 & h22 & h23 & h24 & h25 --> o1 & o2
    
    style input fill:#e1f5ff
    style hidden1 fill:#e8e1ff
    style hidden2 fill:#e1ffe8
    style output fill:#ffe1e1

Multi-Layer Neural Network Architecture

Key insight: Data flows forward through the network, with each layer creating progressively more abstract representations.

Now let’s implement this:

class SimpleNeuralNetwork:
    """A simple feedforward neural network."""
    
    def __init__(self, layer_sizes):
        """
        Args:
            layer_sizes: list of layer sizes, e.g., [3, 8, 5, 2]
                        means 3 inputs, two hidden layers (8 and 5 neurons),
                        and 2 output neurons
        """
        self.layers = []
        for i in range(len(layer_sizes) - 1):
            layer = NeuralLayer(layer_sizes[i], layer_sizes[i + 1])
            self.layers.append(layer)
    
    def forward(self, inputs):
        """Forward pass through all layers."""
        activation = inputs
        
        print("Forward pass through network:")
        print(f"Input shape: {activation.shape}")
        
        for i, layer in enumerate(self.layers):
            # Use ReLU for hidden layers, linear for output
            if i < len(self.layers) - 1:
                activation = layer.forward(activation, activation_fn=relu)
            else:
                activation = layer.forward(activation, activation_fn=lambda x: x)
            
            print(f"After layer {i+1}: {activation.shape}")
        
        return activation

# Create a network: 3 inputs -> 8 hidden -> 5 hidden -> 2 outputs
network = SimpleNeuralNetwork([3, 8, 5, 2])

# Process a batch of data
batch = np.random.randn(4, 3)
output = network.forward(batch)

print(f"\nFinal output:\n{output}")

Forward pass through network:
Input shape: (4, 3)
After layer 1: (4, 8)
After layer 2: (4, 5)
After layer 3: (4, 2)

Final output:
[[ 0.00230489  0.00411068]
 [-0.00171528  0.00183165]
 [ 0.01574223  0.01413083]
 [ 0.00275095  0.00187454]]

What to notice: - Each layer transforms the data: (batch, n_in) -> (batch, n_out) - The output of one layer becomes the input to the next - This creates a series of increasingly abstract representations

Visualizing the Transformation

Let’s see how data is transformed as it flows through the network:

# Create a simpler network for visualization
viz_network = SimpleNeuralNetwork([2, 4, 3, 2])

# Create some structured input data
theta = np.linspace(0, 2*np.pi, 100)
inputs = np.column_stack([np.cos(theta), np.sin(theta)])

# Track activations at each layer
activations = [inputs]
current = inputs

for i, layer in enumerate(viz_network.layers):
    if i < len(viz_network.layers) - 1:
        current = layer.forward(current, activation_fn=relu)
    else:
        current = layer.forward(current, activation_fn=lambda x: x)
    activations.append(current)

# Visualize the transformations
fig, axes = plt.subplots(1, 4, figsize=(16, 4))

for i, (ax, act) in enumerate(zip(axes, activations)):
    if i == 0:
        title = "Input Space"
    elif i == len(activations) - 1:
        title = "Output Space"
    else:
        title = f"Hidden Layer {i}"
    
    # Plot first two dimensions
    ax.scatter(act[:, 0], act[:, 1], c=theta, cmap='viridis', s=20)
    ax.set_title(title, fontsize=12, fontweight='bold')
    ax.set_xlabel('Dimension 1')
    ax.set_ylabel('Dimension 2')
    ax.grid(True, alpha=0.3)
    ax.axis('equal')

plt.tight_layout()
plt.show()

What to notice: Each layer progressively transforms the data, creating new representations. The network learns to map inputs to outputs through these transformations.

---

Part 3: Attention Mechanisms and Transformers

The Limitation of Basic Neural Networks

Traditional feedforward networks process each input independently. But what if relationships between inputs matter? This is where attention mechanisms come in.

Video Resource

For an excellent visual explanation of attention and transformers, watch: - Attention in transformers, visually explained by 3Blue1Brown - Visualizing Attention, a Transformer’s Heart (full explanation)

What is Attention?

Attention allows the network to focus on relevant parts of the input when processing each element. Think of reading a sentence: to understand “it,” you need to look back and find what “it” refers to.

flowchart TB
    subgraph input["Input Sequence"]
        t1["Token 1"]
        t2["Token 2"]
        t3["Token 3"]
        t4["Token 4"]
    end
    
    subgraph qkv["Linear Projections"]
        direction LR
        q["Query (Q)<br/>What I'm looking for"]
        k["Key (K)<br/>What I offer"]
        v["Value (V)<br/>What I return"]
    end
    
    subgraph attention["Attention Computation"]
        similarity["Compute Similarity<br/>Q · Kᵀ"]
        weights["Softmax<br/>(Attention Weights)"]
        output["Weighted Sum<br/>Σ(weights × V)"]
    end
    
    subgraph result["Context-Aware Output"]
        o1["Output 1<br/>(informed by all tokens)"]
        o2["Output 2<br/>(informed by all tokens)"]
        o3["Output 3<br/>(informed by all tokens)"]
        o4["Output 4<br/>(informed by all tokens)"]
    end
    
    t1 & t2 & t3 & t4 --> qkv
    qkv --> similarity
    similarity --> weights
    weights --> output
    output --> o1 & o2 & o3 & o4
    
    style input fill:#e1f5ff
    style qkv fill:#fff3e1
    style attention fill:#f0e1ff
    style result fill:#e1ffe1

Attention Mechanism: Query-Key-Value

Key insight: Each token can “attend to” (look at) all other tokens, deciding which are most relevant for its own representation.

def simple_attention(query, keys, values):
    """
    Simplified attention mechanism.
    
    Args:
        query: what we're looking for (n_queries, d_k)
        keys: what we're comparing against (n_keys, d_k)
        values: what we return (n_keys, d_v)
    
    Returns:
        weighted combination of values
    """
    # Step 1: Compute similarity scores
    # How much does each key match the query?
    scores = np.dot(query, keys.T)  # (n_queries, n_keys)
    
    # Step 2: Convert to attention weights (softmax)
    # Scale by sqrt of dimension for stability
    d_k = keys.shape[1]
    scores = scores / np.sqrt(d_k)
    attention_weights = np.exp(scores) / np.exp(scores).sum(axis=1, keepdims=True)
    
    # Step 3: Weighted sum of values
    output = np.dot(attention_weights, values)  # (n_queries, d_v)
    
    return output, attention_weights

# Example: A simple sequence
# Let's say we have 4 tokens (words), each represented by a 3D vector
sequence = np.array([
    [1.0, 0.0, 0.5],  # Token 0
    [0.5, 1.0, 0.2],  # Token 1
    [0.2, 0.3, 1.0],  # Token 2
    [1.0, 0.5, 0.3],  # Token 3
])

# Let's see what Token 2 attends to
query = sequence[2:3]  # Query is Token 2
keys = sequence        # Keys are all tokens
values = sequence      # Values are all tokens (simplified)

output, attention_weights = simple_attention(query, keys, values)

print("Attention weights for Token 2:")
for i, weight in enumerate(attention_weights[0]):
    print(f"  Token {i}: {weight:.3f}")

print(f"\nOriginal Token 2: {sequence[2]}")
print(f"After attention:  {output[0]}")

Attention weights for Token 2:
  Token 0: 0.238
  Token 1: 0.225
  Token 2: 0.305
  Token 3: 0.231

Original Token 2: [0.2 0.3 1. ]
After attention:  [0.64324521 0.43223908 0.53893462]

What to notice: - The attention weights sum to 1.0 - Token 2 attends most to itself, but also considers other tokens - The output is a weighted combination - it’s been “informed” by the context

Visualizing Attention Patterns

# Compute full attention matrix (all tokens attending to all tokens)
all_queries = sequence
all_keys = sequence
all_values = sequence

outputs, full_attention = simple_attention(all_queries, all_keys, all_values)

# Visualize the attention matrix
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Attention matrix
im = ax1.imshow(full_attention, cmap='viridis', aspect='auto')
ax1.set_title('Attention Matrix\n(Which tokens attend to which)', 
              fontsize=12, fontweight='bold')
ax1.set_xlabel('Key Token')
ax1.set_ylabel('Query Token')
ax1.set_xticks(range(4))
ax1.set_yticks(range(4))
plt.colorbar(im, ax=ax1, label='Attention Weight')

# Add values to cells
for i in range(4):
    for j in range(4):
        text = ax1.text(j, i, f'{full_attention[i, j]:.2f}',
                       ha="center", va="center", color="white", fontweight='bold')

# Show transformation
ax2.bar(range(4), sequence[:, 0], alpha=0.5, label='Original (dim 0)')
ax2.bar(range(4), outputs[:, 0], alpha=0.5, label='After attention (dim 0)')
ax2.set_title('Token Representations\n(First dimension)', 
              fontsize=12, fontweight='bold')
ax2.set_xlabel('Token')
ax2.set_ylabel('Value')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.

What to notice: The attention matrix shows which tokens influence each other. Diagonal values are often high (tokens attend to themselves), but off-diagonal values capture contextual relationships.

Multi-Head Attention

Transformers use multiple attention heads to capture different types of relationships simultaneously.

flowchart TB
    subgraph input_seq["Input Sequence (d_model = 512)"]
        seq["Token 1, Token 2, ..., Token n"]
    end
    
    subgraph linear_proj["Linear Projections"]
        wq["W_Q"]
        wk["W_K"]
        wv["W_V"]
    end
    
    subgraph heads["Split into Multiple Heads (e.g., 8 heads × 64 dims)"]
        direction LR
        head1["Head 1<br/>🔵<br/>Attends to<br/>Syntax"]
        head2["Head 2<br/>🟢<br/>Attends to<br/>Semantics"]
        head3["Head 3<br/>🟡<br/>Attends to<br/>Position"]
        dots["..."]
        head8["Head 8<br/>🔴<br/>Attends to<br/>Context"]
    end
    
    subgraph attention_ops["Parallel Attention Operations"]
        att1["Attention<br/>Computation"]
        att2["Attention<br/>Computation"]
        att3["Attention<br/>Computation"]
        att4["Attention<br/>Computation"]
    end
    
    subgraph concat["Concatenate Heads"]
        combined["Combined Output<br/>(8 heads × 64 = 512 dims)"]
    end
    
    subgraph final["Final Projection"]
        wo["W_O<br/>(512 × 512)"]
        output["Context-Aware<br/>Representations"]
    end
    
    seq --> linear_proj
    linear_proj --> heads
    head1 --> att1
    head2 --> att2
    head3 --> att3
    head8 --> att4
    att1 & att2 & att3 & att4 --> combined
    combined --> wo
    wo --> output
    
    style input_seq fill:#e1f5ff
    style linear_proj fill:#fff3e1
    style heads fill:#f0e1ff
    style attention_ops fill:#ffe1f0
    style concat fill:#e1ffe8
    style final fill:#e1ffe1

Multi-Head Attention Architecture

Key insight: Multiple heads can learn different attention patterns - some might focus on nearby words, others on distant relationships, enabling richer representations.

class MultiHeadAttention:
    """Multi-head attention mechanism."""
    
    def __init__(self, d_model, n_heads):
        """
        Args:
            d_model: dimension of the model (e.g., 512)
            n_heads: number of attention heads (e.g., 8)
        """
        self.n_heads = n_heads
        self.d_model = d_model
        self.d_k = d_model // n_heads  # dimension per head
        
        # Linear projections for Q, K, V
        self.W_q = np.random.randn(d_model, d_model) * 0.1
        self.W_k = np.random.randn(d_model, d_model) * 0.1
        self.W_v = np.random.randn(d_model, d_model) * 0.1
        self.W_o = np.random.randn(d_model, d_model) * 0.1
    
    def split_heads(self, x):
        """Split the last dimension into (n_heads, d_k)."""
        batch_size, seq_len, d_model = x.shape
        # Reshape to (batch_size, seq_len, n_heads, d_k)
        x = x.reshape(batch_size, seq_len, self.n_heads, self.d_k)
        # Transpose to (batch_size, n_heads, seq_len, d_k)
        return x.transpose(0, 2, 1, 3)
    
    def attention(self, q, k, v):
        """Scaled dot-product attention."""
        d_k = q.shape[-1]
        scores = np.matmul(q, k.transpose(0, 1, 3, 2)) / np.sqrt(d_k)
        attention_weights = np.exp(scores) / np.exp(scores).sum(axis=-1, keepdims=True)
        output = np.matmul(attention_weights, v)
        return output, attention_weights
    
    def forward(self, x):
        """
        Forward pass.
        
        Args:
            x: input tensor of shape (batch_size, seq_len, d_model)
        
        Returns:
            output of shape (batch_size, seq_len, d_model)
        """
        batch_size, seq_len, _ = x.shape
        
        # Linear projections
        q = np.dot(x, self.W_q)
        k = np.dot(x, self.W_k)
        v = np.dot(x, self.W_v)
        
        # Split into multiple heads
        q = self.split_heads(q.reshape(batch_size, seq_len, self.d_model))
        k = self.split_heads(k.reshape(batch_size, seq_len, self.d_model))
        v = self.split_heads(v.reshape(batch_size, seq_len, self.d_model))
        
        # Apply attention
        attended, attention_weights = self.attention(q, k, v)
        
        # Concatenate heads
        attended = attended.transpose(0, 2, 1, 3)
        concatenated = attended.reshape(batch_size, seq_len, self.d_model)
        
        # Final linear projection
        output = np.dot(concatenated, self.W_o)
        
        return output, attention_weights

# Create multi-head attention
d_model = 8
n_heads = 2
mha = MultiHeadAttention(d_model, n_heads)

# Process a sequence
batch_size = 1
seq_len = 4
x = np.random.randn(batch_size, seq_len, d_model)

output, attention_weights = mha.forward(x)

print(f"Input shape: {x.shape}")
print(f"Output shape: {output.shape}")
print(f"Attention weights shape: {attention_weights.shape}")
print(f"\nNumber of heads: {n_heads}")
print(f"Each head attends to sequence of length: {seq_len}")

Input shape: (1, 4, 8)
Output shape: (1, 4, 8)
Attention weights shape: (1, 2, 4, 4)

Number of heads: 2
Each head attends to sequence of length: 4

What to notice:

• Each head learns different attention patterns
• Multiple heads capture multiple types of relationships simultaneously
• This is much more powerful than single-head attention

How Attention Changes Encodings

Let’s see how attention modifies representations compared to a simple feedforward layer:

flowchart TB
    subgraph comparison["Processing Paradigms"]
        direction LR
        
        subgraph ff["Feedforward Network"]
            direction TB
            ff_input["Token 1 | Token 2 | Token 3 | Token 4"]
            ff_process["↓ Independent Processing ↓"]
            ff_layer["Layer applies SAME transformation<br/>to each position separately"]
            ff_output["Output 1 | Output 2 | Output 3 | Output 4"]
            ff_note["❌ No communication between positions"]
            
            ff_input --> ff_process
            ff_process --> ff_layer
            ff_layer --> ff_output
            ff_output --> ff_note
        end
        
        subgraph attn["Attention Network"]
            direction TB
            attn_input["Token 1 | Token 2 | Token 3 | Token 4"]
            attn_process["↓ Context-Aware Processing ↓"]
            attn_layer["Each position looks at ALL positions<br/>Weighted combination based on relevance"]
            attn_output["Output 1 | Output 2 | Output 3 | Output 4"]
            attn_note["✅ Each output informed by full context"]
            
            attn_input --> attn_process
            attn_process --> attn_layer
            attn_layer --> attn_output
            attn_output --> attn_note
        end
    end
    
    subgraph examples["Real-World Example"]
        direction LR
        sentence["'The animal didn't cross the street because it was too tired'"]
        ff_ex["Feedforward: 'it' processed in isolation<br/>❌ Can't determine if 'it' = animal or street"]
        attn_ex["Attention: 'it' attends to all words<br/>✅ Learns 'it' → 'animal' (via context)"]
        
        sentence --> ff_ex
        sentence --> attn_ex
    end
    
    style ff fill:#ffe1e1
    style attn fill:#e1ffe1
    style ff_note fill:#ffcccc
    style attn_note fill:#ccffcc
    style examples fill:#fff9e1

Feedforward vs Attention: The Key Difference

Why this is revolutionary: Attention allows the network to dynamically route information based on context, rather than applying fixed transformations. This is essential for understanding language, time series, and sequential data where relationships between elements matter.

# Create sample sequence data
np.random.seed(42)
seq_length = 6
d_model = 8

# Input sequence
input_seq = np.random.randn(1, seq_length, d_model)

# Option 1: Process with feedforward layer (no attention)
ff_layer = NeuralLayer(d_model, d_model)
ff_output = ff_layer.forward(input_seq.reshape(-1, d_model), activation_fn=relu)
ff_output = ff_output.reshape(1, seq_length, d_model)

# Option 2: Process with attention
mha = MultiHeadAttention(d_model, n_heads=2)
attn_output, attn_weights = mha.forward(input_seq)

# Visualize the difference
fig, axes = plt.subplots(1, 3, figsize=(16, 4))

# Input
im0 = axes[0].imshow(input_seq[0].T, cmap='RdBu', aspect='auto', vmin=-2, vmax=2)
axes[0].set_title('Input Sequence', fontsize=12, fontweight='bold')
axes[0].set_xlabel('Position in Sequence')
axes[0].set_ylabel('Feature Dimension')
plt.colorbar(im0, ax=axes[0])

# Feedforward output
im1 = axes[1].imshow(ff_output[0].T, cmap='RdBu', aspect='auto', vmin=-2, vmax=2)
axes[1].set_title('After Feedforward Layer\n(No attention)', fontsize=12, fontweight='bold')
axes[1].set_xlabel('Position in Sequence')
axes[1].set_ylabel('Feature Dimension')
plt.colorbar(im1, ax=axes[1])

# Attention output
im2 = axes[2].imshow(attn_output[0].T, cmap='RdBu', aspect='auto', vmin=-2, vmax=2)
axes[2].set_title('After Multi-Head Attention\n(Context-aware)', fontsize=12, fontweight='bold')
axes[2].set_xlabel('Position in Sequence')
axes[2].set_ylabel('Feature Dimension')
plt.colorbar(im2, ax=axes[2])

plt.tight_layout()
plt.show()

# Show how representations relate to each other
print("\nCosine similarity between position encodings:\n")
print("Feedforward (independent processing):")
ff_norm = ff_output[0] / np.linalg.norm(ff_output[0], axis=1, keepdims=True)
ff_similarity = np.dot(ff_norm, ff_norm.T)
print(f"Average off-diagonal similarity: {(ff_similarity.sum() - seq_length) / (seq_length * (seq_length - 1)):.3f}")

print("\nAttention (context-aware processing):")
attn_norm = attn_output[0] / np.linalg.norm(attn_output[0], axis=1, keepdims=True)
attn_similarity = np.dot(attn_norm, attn_norm.T)
print(f"Average off-diagonal similarity: {(attn_similarity.sum() - seq_length) / (seq_length * (seq_length - 1)):.3f}")

Cosine similarity between position encodings:

Feedforward (independent processing):
Average off-diagonal similarity: 0.421

Attention (context-aware processing):
Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.
Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.
Average off-diagonal similarity: 0.998

Why this matters:

• Feedforward layers process each position independently - no position “knows” about others
• Attention layers mix information across positions - each position is informed by context
• This context-awareness is crucial for sequential data like language, time series, or video

---

Part 4: Putting It All Together - The Transformer Block

A complete transformer block combines attention with feedforward layers:

flowchart TB
    input["Input<br/>(Sequence of Embeddings)"]
    
    subgraph mha_block["Multi-Head Attention Block"]
        mha["Multi-Head<br/>Attention"]
        add1["Add"]
        norm1["Layer Norm"]
    end
    
    subgraph ff_block["Feedforward Block"]
        ff1["Linear<br/>(expand)"]
        relu["ReLU"]
        ff2["Linear<br/>(project)"]
        add2["Add"]
        norm2["Layer Norm"]
    end
    
    output["Output<br/>(Enriched Representations)"]
    
    input --> mha
    input -.->|"Residual<br/>Connection"| add1
    mha --> add1
    add1 --> norm1
    
    norm1 --> ff1
    ff1 --> relu
    relu --> ff2
    norm1 -.->|"Residual<br/>Connection"| add2
    ff2 --> add2
    add2 --> norm2
    
    norm2 --> output
    
    style input fill:#e1f5ff
    style mha_block fill:#f0e1ff
    style ff_block fill:#ffe8e1
    style output fill:#e1ffe1
    style mha fill:#d8b3ff
    style add1 fill:#ffd8b3
    style add2 fill:#ffd8b3

Complete Transformer Block Architecture

Key components:

1. Multi-Head Attention: Mix information across positions (context)
2. Residual Connection: Add input back to help gradient flow
3. Layer Normalization: Stabilize training
4. Feedforward Network: Transform features independently
5. Another Residual Connection: More gradient flow

This pattern repeats for each transformer layer in a model!

class TransformerBlock:
    """A single transformer block with attention and feedforward."""
    
    def __init__(self, d_model, n_heads, d_ff):
        """
        Args:
            d_model: model dimension
            n_heads: number of attention heads
            d_ff: feedforward network dimension
        """
        self.attention = MultiHeadAttention(d_model, n_heads)
        self.ff1 = NeuralLayer(d_model, d_ff)
        self.ff2 = NeuralLayer(d_ff, d_model)
        
    def forward(self, x):
        """
        Forward pass through transformer block.
        
        Args:
            x: input of shape (batch_size, seq_len, d_model)
        
        Returns:
            output of same shape as input
        """
        batch_size, seq_len, d_model = x.shape
        
        # Step 1: Multi-head attention
        attn_out, attn_weights = self.attention.forward(x)
        
        # Step 2: Add & Norm (simplified - just add)
        x = x + attn_out
        
        # Step 3: Feedforward network
        # Reshape for layer processing
        x_flat = x.reshape(-1, d_model)
        ff_out = self.ff1.forward(x_flat, activation_fn=relu)
        ff_out = self.ff2.forward(ff_out, activation_fn=lambda x: x)
        ff_out = ff_out.reshape(batch_size, seq_len, d_model)
        
        # Step 4: Add & Norm (simplified - just add)
        output = x + ff_out
        
        return output, attn_weights

# Create and test a transformer block
transformer = TransformerBlock(d_model=8, n_heads=2, d_ff=16)

# Process a sequence
input_seq = np.random.randn(1, 6, 8)
output, attention = transformer.forward(input_seq)

print(f"Input shape: {input_seq.shape}")
print(f"Output shape: {output.shape}")
print("\nTransformer block completed:")
print("  1. Multi-head attention (context mixing)")
print("  2. Residual connection")
print("  3. Feedforward network (feature transformation)")
print("  4. Residual connection")

Input shape: (1, 6, 8)
Output shape: (1, 6, 8)

Transformer block completed:
  1. Multi-head attention (context mixing)
  2. Residual connection
  3. Feedforward network (feature transformation)
  4. Residual connection

What to notice: The transformer combines two key ideas:

1. Attention: Mix information across sequence positions (capture context)
2. Feedforward: Transform features independently at each position (extract patterns)
3. Residual connections: Add the input back to help gradients flow

---

Summary: The Neural Network Hierarchy

Let’s recap the progressive complexity:

flowchart TD
    subgraph level1["Level 1: Single Neuron"]
        n1["Weighted Sum<br/>+ Activation<br/>────<br/>z = Σ(wᵢxᵢ) + b<br/>output = f(z)"]
    end
    
    subgraph level2["Level 2: Neural Layer"]
        layer["Vectorized Operations<br/>────<br/>Multiple neurons in parallel<br/>Matrix: (n_in, n_out)"]
    end
    
    subgraph level3["Level 3: Deep Network"]
        deep["Stacked Layers<br/>────<br/>Input → Hidden₁ → Hidden₂ → Output<br/>Hierarchical features"]
    end
    
    subgraph level4a["Level 4a: Attention Mechanism"]
        attention["Query-Key-Value<br/>────<br/>Context-aware weighting<br/>Tokens attend to each other"]
    end
    
    subgraph level4b["Level 4b: Multi-Head Attention"]
        multihead["Parallel Attention Heads<br/>────<br/>Multiple relationship types<br/>8 heads × 64 dims = 512 dims"]
    end
    
    subgraph level5["Level 5: Transformer Block"]
        transformer["Complete Architecture<br/>────<br/>Multi-Head Attention<br/>+ Residual Connection<br/>+ Feedforward Network<br/>+ Residual Connection"]
    end
    
    subgraph level6["Level 6: Full Transformer"]
        full["Stack of N Blocks<br/>────<br/>Block₁ → Block₂ → ... → Blockₙ<br/>+ Positional Encoding<br/>+ Output Head"]
    end
    
    level1 -.->|"Parallelize"| level2
    level2 -.->|"Stack"| level3
    level3 -.->|"Add Context<br/>Awareness"| level4a
    level4a -.->|"Multiple<br/>Heads"| level4b
    level4b -.->|"+ Feedforward<br/>+ Residuals"| level5
    level5 -.->|"Stack<br/>Layers"| level6
    
    capability1["Independent<br/>Processing"] -.-> level1
    capability2["Batch<br/>Processing"] -.-> level2
    capability3["Hierarchical<br/>Features"] -.-> level3
    capability4["Sequential<br/>Dependencies"] -.-> level4a
    capability5["Rich<br/>Relationships"] -.-> level4b
    capability6["Stable Deep<br/>Learning"] -.-> level5
    capability7["Complex<br/>Understanding"] -.-> level6
    
    style level1 fill:#e1f5ff
    style level2 fill:#e8f0ff
    style level3 fill:#e8e5ff
    style level4a fill:#f0e1ff
    style level4b fill:#f8e1ff
    style level5 fill:#ffe1f0
    style level6 fill:#ffe1e1
    
    style capability1 fill:#fff9e1
    style capability2 fill:#fff9e1
    style capability3 fill:#fff9e1
    style capability4 fill:#fff9e1
    style capability5 fill:#fff9e1
    style capability6 fill:#fff9e1
    style capability7 fill:#fff9e1

The Complete Neural Network Hierarchy: From Neurons to Transformers

Level 1: Single Neuron

• Computes weighted sum of inputs
• Applies activation function (e.g., ReLU)
• Transforms: scalar inputs → scalar output
• Key insight: Non-linearity enables learning complex patterns

Level 2: Layer of Neurons

• Multiple neurons computed in parallel (vectorized)
• Matrix multiplication: efficient batch processing
• Transforms: vector input → vector output
• Key insight: Multiple features extracted simultaneously

Level 3: Multi-Layer Network

• Stack layers to create deep representations
• Each layer builds on previous abstractions
• Transforms: input space → hidden spaces → output space
• Key insight: Depth creates hierarchical features

Level 4: Attention & Transformers

• Attention: dynamically weight inputs based on context
• Multi-head: capture multiple relationship types
• Transformer: attention + feedforward with residual connections
• Key insight: Context-aware processing for sequential data

The Key Difference

Traditional Neural Networks: Process each input independently

• Good for: images, tabular data where position doesn’t matter

Transformers with Attention: Process inputs in context of each other

• Good for: language, time series, any sequential data
• Revolution: Enables models like GPT, BERT, and modern GFMs

---

Interactive Exploration

Try This Yourself

Experiment with the code above:

1. Activation functions: Change the activation in single_neuron() and observe output changes
2. Layer sizes: Modify layer_sizes in SimpleNeuralNetwork - what happens with very wide or very deep networks?
3. Attention heads: Increase n_heads in MultiHeadAttention - do patterns change?
4. Sequence length: Use longer sequences in attention examples - observe attention patterns

These experiments will deepen your intuition about how neural networks transform data.

---

Further Resources

Video Explanations (3Blue1Brown)

1. Neural Networks - Core concepts
2. Gradient Descent - How networks learn
3. Backpropagation - How gradients flow
4. Attention & Transformers - Modern architecture

Key Papers

• “Attention Is All You Need” (Vaswani et al., 2017) - The original transformer paper
• “Deep Residual Learning” (He et al., 2015) - Residual connections
• “Understanding Deep Learning Requires Rethinking Generalization” (Zhang et al., 2017)

Next Steps in This Course

• Week 2: Spatial-temporal attention for geospatial data
• Week 3: Vision Transformers adapted for satellite imagery
• Week 4: Pretraining strategies (masked autoencoders)

---

This explainer is designed to build progressive understanding. Each section assumes you’ve understood the previous ones. Take time to run the code, observe the outputs, and experiment!