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Artificial Neural Networks (ANNs) are powerful computational models inspired by the human brain, and their strength lies in the mathematics that governs how they learn and make predictions. At their core, ANNs use linear algebra to represent data as vectors and matrices, and apply weighted transformations to propagate information through layers. Non-linear activation functions introduce complexity, enabling networks to learn relationships beyond simple linear patterns.Learning in ANNs is driven by calculus, specifically gradient-based optimization. Using backpropagation, networks compute partial derivatives of the loss function with respect to each weight, adjusting parameters to minimize error. Probability and statistics further support ANN behavior by defining loss functions, modeling uncertainty, and improving generalization. Overall, the mathematics of ANNs forms the foundation for training stable, accurate, and scalable deep learning models.

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🛠️ Tech Stack & Tools

Category	Tool	Badge	Purpose
Language	Python 3.9+		Primary programming language
Core Library	NumPy		Numerical computing & matrix operations
Visualization	Matplotlib		Data visualization & plotting
ML Utilities	Scikit-Learn		Dataset loading & preprocessing
Notebook	Jupyter		Interactive development environment
Editor	VS Code		Code editor & IDE
Version Control	Git		Source code management
Repository	GitHub		Code hosting & collaboration
Framework	TensorFlow		Benchmarking & comparison only

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Project Structure

#	Project	Folder	Key Concepts
01	Perceptron Learning Rule	01_perceptron_learning	Linear separability, weight updates
02	XOR with MLP	02_xor_mlp	Non-linearity, backpropagation
03	MNIST Digit Recognition	03_mnist_digit_recognition	Multi-class classification, softmax
04	Neural Network Visualizer	04_nn_visualizer	Training dynamics, weight evolution
05	Custom Dataset ANN	05_custom_dataset_ann	Tabular data, label encoding
06	Loss Surface Visualization	06_loss_landscape	Loss contours, optimization geometry
07	Backpropagation Simulator	07_backprop_simulator	Chain rule, matrix calculus
08	Activation Function Analysis	08_activation_function_analysis	ReLU vs. Sigmoid vs. Tanh
09	Dropout Regularization	09_dropout_regularization	Overfitting prevention
10	Time Series Forecasting	10_time_series_ann	Sliding window, ANN regression

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Architecture Overview

                           ┌────────────────────────┐
                           │ Mathematics of ANN     │
                           └─────────────┬──────────┘
                                         │
        ┌────────────────────────────────┼─────────────────────────────────┐
        │                                │                                 │
        ▼                                ▼                                 ▼
┌────────────────┐             ┌────────────────────┐             ┌────────────────────┐
│  Core Building │             │ Training &         │             │ Advanced Concepts  │
│  Blocks        │             │ Optimization       │             │ & Experiments      │
└───────┬────────┘             └─────────┬──────────┘             └─────────┬──────────┘
        │                                │                                  │
        ▼                                ▼                                  ▼
┌──────────────────┐          ┌──────────────────────┐          ┌──────────────────────────┐
│ • Perceptron     │          │ • Gradient Descent   │          │ • Backprop Simulator     │
│ • Activation     │          │ • Loss Functions     │          │ • Dropout Regularization │
│   Functions      │          │ • Loss Landscape Viz │          │ • Time Series Forecast   │
└──────────────────┘          └──────────────────────┘          │ • MNIST Recognition      │
                                                                └──────────────────────────┘

Quick Start

Prerequisites

Recommended Knowledge:

• Basic linear algebra and calculus
• Python programming (NumPy basics)
• Understanding of gradient descent

Installation

# Clone the repository
git clone https://github.com/arun-techverse/neural-networks-from-scratch-math-projects.git
cd neural-networks-from-scratch-math-projects

# Create virtual environment (recommended)
python -m venv venv
source venv/bin/activate  # On Windows: venv\Scripts\activate

# Install dependencies
pip install -r requirements.txt

# Launch Jupyter Notebook
jupyter notebook

Dependencies

numpy>=1.21.0
matplotlib>=3.4.0
scikit-learn>=1.0.0
tensorflow>=2.8.0  # For comparison only
jupyter>=1.0.0

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Activation Functions

Activation functions introduce non-linearity into neural networks, enabling them to learn complex patterns. Without them, networks would behave like linear regression regardless of depth.

Common Activation Functions

Function	Formula	Range	Use Case
Sigmoid	$f(x) = \frac{1}{1 + e^{-x}}$	(0, 1)	Binary classification output
Tanh	$f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$	(-1, 1)	Hidden layers (zero-centered)
ReLU	$f(x) = \max(0, x)$	[0, ∞)	Hidden layers (most popular)
Softmax	$P(y=i) = \frac{e^{z_i}}{\sum_{j} e^{z_j}}$	(0, 1)	Multi-class classification

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