DANIEL MARIN

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Understanding Neural Networks

A deep dive into the fundamentals of neural networks and their mathematical foundations

•1 min read

Neural networks are the backbone of modern artificial intelligence. Let's explore how they work from the ground up.

The Basic Neuron

A single neuron performs a simple computation:

import numpy as np

def neuron(inputs, weights, bias):
    """
    Simple neuron computation
    """
    weighted_sum = np.dot(inputs, weights) + bias
    return activation(weighted_sum)

def sigmoid(x):
    """Sigmoid activation function"""
    return 1 / (1 + np.exp(-x))

Mathematical Foundation

The output of a neuron can be expressed as:

y=f(∑i=1nwixi+b)y = f\left(\sum_{i=1}^{n} w_i x_i + b\right)

Where:

  • xix_i are the inputs
  • wiw_i are the weights
  • bb is the bias
  • ff is the activation function

Forward Propagation

In a multi-layer network, the output of one layer becomes the input to the next:

a(l)=f(W(l)a(l−1)+b(l))a^{(l)} = f(W^{(l)} a^{(l-1)} + b^{(l)})

This process continues through all layers until we reach the output.

Backpropagation

The magic happens during training through backpropagation, which uses the chain rule to compute gradients:

∂L∂wij(l)=∂L∂aj(l)⋅∂aj(l)∂zj(l)⋅∂zj(l)∂wij(l)\frac{\partial L}{\partial w_{ij}^{(l)}} = \frac{\partial L}{\partial a_j^{(l)}} \cdot \frac{\partial a_j^{(l)}}{\partial z_j^{(l)}} \cdot \frac{\partial z_j^{(l)}}{\partial w_{ij}^{(l)}}

Stay tuned for more posts on advanced neural network architectures!