  We suggest a general approach to quantification of different forms of
aleatoric uncertainty in regression tasks performed by artificial neural
networks. It is based on the simultaneous training of two neural networks with
a joint loss function and a specific hyperparameter $\lambda>0$ that allows for
automatically detecting noisy and clean regions in the input space and
controlling their {\em relative contribution} to the loss and its gradients.
After the model has been trained, one of the networks performs predictions and
the other quantifies the uncertainty of these predictions by estimating the
locally averaged loss of the first one. Unlike in many classical uncertainty
quantification methods, we do not assume any a priori knowledge of the ground
truth probability distribution, neither do we, in general, maximize the
likelihood of a chosen parametric family of distributions. We analyze the
learning process and the influence of clean and noisy regions of the input
space on the loss surface, depending on $\lambda$. In particular, we show that
small values of $\lambda$ increase the relative contribution of clean regions
to the loss and its gradients. This explains why choosing small $\lambda$
allows for better predictions compared with neural networks without uncertainty
counterparts and those based on classical likelihood maximization. Finally, we
demonstrate that one can naturally form ensembles of pairs of our networks and
thus capture both aleatoric and epistemic uncertainty and avoid overfitting.
