Deep Learning Multi-Layer Neural Network Visualizer

Artificial Intelligence & Deep Learning

Demystifying Deep Learning & Backpropagation

A Visual & Mathematical Deep Dive into Multi-Layered Neural Networks

In the study of modern Artificial Intelligence—specifically within frameworks such as the EXIN BCS Generative AI Award syllabus—Deep Learning (DL) stands out as a critical subfield of Machine Learning. Unlike traditional rule-based or shallow statistical systems, deep learning utilizes multi-layered artificial neural networks capable of modeling complex, non-linear structures in massive datasets.

1. What is Deep Learning?

At its core, deep learning stacks multiple artificial neural layers between raw input data and final predictions:

• Input Layer: Receives raw features (e.g., pixel intensities, token embeddings, sensor readings).
• Hidden Layers: Perform mathematical transformations (weighted sums + non-linear activations) to extract hierarchical features.
• Output Layer: Generates the target prediction (e.g., classification labels or continuous numerical values).

2. Interactive Visualizer: Single-File HTML5 CodePen Project

To visualize signal propagation across multiple stacked hidden layers, here is a complete, self-contained HTML5 & JavaScript project that you can embed or run directly in CodePen:

3. Understanding Backpropagation

If the forward pass is how a network makes a guess, backpropagation (backward propagation of errors) is how it learns from its mistakes.

The 4-Stage Learning Cycle

1. Forward Pass: Inputs move forward computing weighted sum z = ∑(w·x) + b and activations a = σ(z).
2. Loss Calculation: Compare network prediction ŷ against target output y via Loss Function L.
3. Backward Pass (Chain Rule): Calculate derivative ∂L / ∂w from the output layer backwards.
4. Weight Update: Adjust parameters via learning rate η: w_new = w_old - η · (∂L / ∂w).

4. Step-by-Step Mathematical Example

Let's calculate one complete update step for a single neuron using real numbers:

Setup Parameters:

• Input (x) = 2.0 | Target (y) = 1.0
• Initial Weight (w) = 0.8 | Bias (b) = 0.1
• Learning Rate (η) = 0.5

Step 1: Forward Pass & Loss Calculation

z = (0.8 · 2.0) + 0.1 = 1.7
ŷ = σ(1.7) = 1 / (1 + e⁻¹˙⁷) ≈ 0.8455
Loss (L) = ½ (0.8455 - 1.0)² ≈ 0.0119

Step 2: Chain Rule Gradient Derivation

∂L / ∂ŷ = (0.8455 - 1.0) = -0.1545
∂ŷ / ∂z = 0.8455 · (1 - 0.8455) ≈ 0.1306
∂z / ∂w = x = 2.0
∂L / ∂w = (-0.1545) · (0.1306) · (2.0) ≈ -0.0404

Step 3: Weight Adjustment

w_new = 0.8 - (0.5 · -0.0404) = 0.8202

5. Summary of Results

Metric	Before Pass	After Pass (w = 0.8202)
Weight (w)	0.8000	0.8202
Prediction (ŷ)	0.8455	0.8508 (Closer to target 1.0)
Loss (L)	0.0119	0.0110 (Decreased)