Uncertainty heads for NNUE

So the normal NNUE loss is this:

$$t = \lambda z + (1-\lambda)\,\sigma\!\left(\tfrac{v}{400}\right)$$

$$v = \mathtt{NETWORK}(x)$$

$$\mathcal{L} = \underbrace{(v - t)^2}_{\text{MSE}}$$

The idea is to instead have the network produce two outputs:

$$v, u = \mathtt{NETWORK}(x)$$

Then make the loss encompass both minimising MSE and predicting it:

$$\begin{aligned} \text{MSE} = (v - t)^2 \\ \mathcal{L} = \text{MSE} + (u - \text{MSE})^2 \end{aligned}$$