Sample Example: Backpropagation Example:

The following imports are require for running the script in Python

• import numpy as np
• import matplotlib.pyplot as plt

Step 1: Initialize Weights and Inputs

$$ w_{1} = 0.15, \quad w_{2} = 0.20, \quad w_{3} = 0.25, \quad w_{4} = 0.30 $$ $$ w_{5} = 0.40, \quad w_{6} = 0.45 $$ $$ x_{1} = 0.05, \quad x_{2} = 0.10 $$

Step 2: Forward Pass (Calculate Activation)

$$ h_{1} = \sigma(w_{1}x_{1} + w_{2}x_{2}) = \sigma(0.0175) $$ $$ h_{2} = \sigma(w_{3}x_{1} + w_{4}x_{2}) = \sigma(0.0275) $$ $$ y = \sigma(w_{5}h_{1} + w_{6}h_{2}) = \sigma(0.75) $$

Step 3: Calculate Loss (Mean Squared Error)

$$ L = \frac{1}{2} (0.85 - 0.75)^2 = 0.005 $$

Step 4: Backpropagate Error

$$ \delta_{\text{output}} = (t - y) \cdot \sigma'(z_{\text{output}}) $$ $$ \delta_{\text{output}} = (0.85 - 0.75) \cdot 0.18 = 0.018 $$

Step 5: Update Weights

$$ w_{5} = w_{5} + \eta \cdot \delta_{\text{output}} \cdot h_{1} $$ $$ w_{6} = w_{6} + \eta \cdot \delta_{\text{output}} \cdot h_{2} $$

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