To find the total number of trainable parameters in the neural network, we need to consider the parameters for each layer in the network:

1. First Recurrent Layer:

• Input Size: (3,1) – 3 features.
• Number of Neurons: 6.

In a recurrent layer, each neuron has weights associated with the input and weights associated with the previous hidden state, along with a bias term.

• Weights for Input to Neuron: For each of the 6 neurons, the input has 3 features. Thus, there are $3 \times 6 = 18$ parameters.
• Weights for Recurrent Connection: Each neuron also has a connection to the previous hidden state of the same layer. Since there are 6 neurons, each with 6 connections (from the previous state), there are $6 \times 6 = 36$ parameters.
• Biases: Each of the 6 neurons has 1 bias parameter, so there are 6 bias parameters.

Total Parameters for the First Recurrent Layer:
$18 \text{ (input weights)} + 36 \text{ (recurrent weights)} + 6 \text{ (biases)} = 60$ parameters.

2. Second Recurrent Layer:

• Input Size: 6 (output from the first layer).

• Number of Neurons: 6.

• Weights for Input to Neuron: The input to this layer comes from the 6 neurons in the first layer. Thus, there are $6 \times 6 = 36$ parameters.

• Weights for Recurrent Connection: Similar to the first layer, the recurrent connections involve 6 neurons, so there are $6 \times 6 = 36$ parameters.

• Biases: Each of the 6 neurons has 1 bias parameter, so there are 6 bias parameters.

Total Parameters for the Second Recurrent Layer:
$36 \text{ (input weights)} + 36 \text{ (recurrent weights)} + 6 \text{ (biases)} = 78$ parameters.

3. Dense Output Layer:

• Input Size: 6 (output from the second recurrent layer).

• Output Size: Scalar (1).

• Weights: There are 6 weights (one for each neuron in the second recurrent layer to the output).

• Bias: There is 1 bias parameter for the output neuron.

Total Parameters for the Dense Layer:
$6 \text{ (weights)} + 1 \text{ (bias)} = 7$ parameters.

Total Number of Trainable Parameters:

$60 \text{ (First Recurrent Layer)} + 78 \text{ (Second Recurrent Layer)} + 7 \text{ (Dense Layer)} = 145 \text{ parameters.}$

So, the total number of trainable parameters in the network is 145.

Would you like more details or have any questions?

Here are some related questions to expand on this:

1. How do recurrent layers differ from dense layers in terms of parameter calculation?
2. What are the advantages of using recurrent layers in neural networks?
3. Can you explain the role of bias parameters in a neural network?
4. How does the size of the input affect the number of trainable parameters in a recurrent layer?
5. What are some common applications of recurrent neural networks?

Tip: When calculating the number of parameters in neural networks, always account for both the weights and biases in each layer.