Mathematics for NN

1. Introduction to the Mathematics of Neural Networks

1.1. Why is mathematics important?

1.2. The connection between mathematics and programming in neural networks

2. Linear and Nonlinear Activation Functions

2.1. Linear functions

2.2. Nonlinear functions (ReLU, Leaky ReLU, Tanh, etc.)

2.3. Graphical analysis of activation functions (Sigmoid, ReLU, Softmax)

2.4 Loss Functions (Cross-Entropy, Mean Squared Error)

3. Gradient Matrices and Backpropagation

3.1. The basic idea of backpropagation (Backpropagation and Weight Optimization)

3.2. Gradient computation using the chain rule

3.3. Examples in PHP

4. Regularization in Neural Networks

4.1 What is overfitting?

4.2. Regularization methods (L1, L2, Dropout)

4.3. Mathematical justification for regularization

5. Numerical Methods and Computation

5.1. Numerical stability in neural networks

5.2. The concept of vanishing and exploding gradients

5.3. Practical techniques for numerical optimization

6. Entropy and Cross-Entropy

6.1 The concept of entropy

6.2. Cross-entropy loss function

6.3. Implementation of the loss function in PHP

7. Conclusion: Combining Mathematics and Programming

7.1. How mathematics aids in building efficient neural networks

7.2. Recommendations for further study