Can you provide an example of a Simple neural network class in pure JavaScript with no libraries.
Sure! Here's an example of a simple neural network class in pure JavaScript:

```javascript
class NeuralNetwork {
  constructor(input_nodes, hidden_nodes, output_nodes, learning_rate) {
    this.input_nodes = input_nodes;
    this.hidden_nodes = hidden_nodes;
    this.output_nodes = output_nodes;
    this.learning_rate = learning_rate;

    // Initialize weights
    this.weights_ih = new Matrix(this.hidden_nodes, this.input_nodes);
    this.weights_ho = new Matrix(this.output_nodes, this.hidden_nodes);
    this.weights_ih.randomize();
    this.weights_ho.randomize();

    // Initialize biases
    this.bias_h = new Matrix(this.hidden_nodes, 1);
    this.bias_o = new Matrix(this.output_nodes, 1);
    this.bias_h.randomize();
    this.bias_o.randomize();
  }

  feedforward(input_array) {
    // Generate hidden layer
    let inputs = Matrix.fromArray(input_array);
    let hidden = Matrix.multiply(this.weights_ih, inputs);
    hidden.add(this.bias_h);
    hidden.map(sigmoid);

    // Generate output layer
    let output = Matrix.multiply(this.weights_ho, hidden);
    output.add(this.bias_o);
    output.map(sigmoid);

    return output.toArray();
  }

  train(input_array, target_array) {
    // Feedforward
    let inputs = Matrix.fromArray(input_array);
    let hidden = Matrix.multiply(this.weights_ih, inputs);
    hidden.add(this.bias_h);
    hidden.map(sigmoid);

    let outputs = Matrix.multiply(this.weights_ho, hidden);
    outputs.add(this.bias_o);
    outputs.map(sigmoid);

    // Backpropagation
    let targets = Matrix.fromArray(target_array);
    let output_errors = Matrix.subtract(targets, outputs);
    let hidden_errors = Matrix.multiply(Matrix.transpose(this.weights_ho), output_errors);

    // Update weights and biases
    let gradients = Matrix.map(outputs, dsigmoid);
    gradients.multiply(output_errors);
    gradients.multiply(this.learning_rate);

    let hidden_T = Matrix.transpose(hidden);
    let weights_ho_deltas = Matrix.multiply(gradients, hidden_T);
    this.weights_ho.add(weights_ho_deltas);
    this.bias_o.add(gradients);

    let hidden_gradients = Matrix.map(hidden, dsigmoid);
    hidden_gradients.multiply(hidden_errors);
    hidden_gradients.multiply(this.learning_rate);

    let inputs_T = Matrix.transpose(inputs);
    let weights_ih_deltas = Matrix.multiply(hidden_gradients, inputs_T);
    this.weights_ih.add(weights_ih_deltas);
    this.bias_h.add(hidden_gradients);
  }
}

function sigmoid(x) {
  return 1 / (1 + Math.exp(-x));
}

function dsigmoid(y) {
  return y * (1 - y);
}
```



This example creates a `NeuralNetwork` class that takes in the number of input nodes, hidden nodes, output nodes, and learning rate as constructor arguments. It initializes weights and biases randomly, and provides methods for feedforward and training using backpropagation.

This implementation uses a `Matrix` class to perform matrix operations, which is not a built-in JavaScript feature. You can define the `Matrix` class yourself or use an existing library like `mathjs`.
What loss function does this example use? I don't see any.
You're right, the code doesn't explicitly define a loss function. However, it implicitly uses mean squared error (MSE) as the loss function during the backpropagation step.

The MSE loss function is defined as:

`L = (1 / N) * Σ(target_i - output_i)^2`

where N is the number of output nodes, target_i is the target value for the i-th output node, and output_i is the actual output value for the i-th output node.

In the train function, the following line calculates the output_errors, which is the difference between the target values and the actual output values:

```javascript
let output_errors = Matrix.subtract(targets, outputs);
```

These output_errors are used to update the weights and biases, effectively minimizing the MSE loss. The code doesn't calculate the actual MSE loss value, but the backpropagation process implicitly optimizes the network to minimize this loss function.

Let me know if there's anything else I can clarify for you.