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The architecture of a neural network, including the number of neurons and layers, depends on the specific problem we are trying to solve. There's no one-size-fits-all answer, and it's not necessarily a solved problem, that is, nobody knows the right answer; it often involves a combination of domain knowledge, experimentation, and tuning.

Some general considerations are:

1. Problem Complexity: More complex problems might require larger networks with more neurons and deeper architectures.

2. Data Availability: The amount of data we have plays a role. Larger datasets may benefit from deeper networks, but if we have a small dataset, a simpler network might be less prone to overfitting.

3. Train, validate and test with multiple architectures. After the tests, then we can decide which one is the best.
4. Computational Resources: Deeper networks with more neurons require more computational power and memory.
5. Computational Efficiency: In some cases, we might prioritize a smaller and more efficient model, especially for deployment on resource-constrained devices.
6. Overfitting and Regularization: Deeper networks are more prone to overfitting, so we may need to use techniques like dropout or regularization to prevent this.
7. Nature of Data: Understanding the nature of the data, such as its structure and patterns, can guide our choice of architecture.
8. Experimentation: It's often necessary to experiment with different architectures to see what works best for our specific problem.
9. Transfer Learning: For some tasks, using a pre-trained model (transfer learning) might be more effective than training a model from scratch.
10. Activation Functions: The choice of activation functions in each layer can also impact the network's performance.

Figure 3724 shows a single neuron with its connections. The green ball represents the activation function, a = σ. Here, z is given by,

          z = w₁x₁+ w₂x₂ + ... + w_nx_n ------------------------------------------ [3724a]

Figure 3724. A single neuron with its connections (Code).

Equation 3724a is a linear combination, where z is a weighted sum of n variables (x₁, x₂, ..., x_n) with corresponding weights (w₁, w₂, ..., w_n). As the number of variables (n) increases, the weights (w_i) need to be smaller to prevent the output (z) from becoming too large, in order to prevent from vanishing and exploding gradients. The appropriate w_i should be,

          ---------------------------------------------- [3724b]

Therefore, weight initialization (choosing appropriate initial values for weights) is crucial in mitigating the issues of vanishing and exploding gradients. The idea is that if weights are too large, gradients can explode; if they are too small, gradients can vanish.

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