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I have a regression task where all my training data that I want to predict is of the form ($y$, $\sigma$), where $\sigma$ is the Gaussian noise corresponding to $y$, i.e. I want to be able to predict both $y$ and its corresponding uncertainty. Are there implementations of neural networks that can account for uncertainty in training such that I can predict $y$ and $\sigma$?

For example, is it valid to simply learn the $\sigma$ directly and treat it on the same footing as y when training the neural network? Should there be any kind of under- or over-sampling depending on the magnitude of σ (e.g. under-sample data with large $\sigma$)? I came across a couple papers on the topic, e.g. "Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles" by Lakshminarayanan et al., but further insight on practical application of deep learning on training data with provided uncertainties would be greatly appreciated. Any references to existing packages/code examples tackling this problem would be very helpful.

(Note: the exact structure of my training data involves about $N = 1 000 000$ samples of training data. Also, there will be about 100 inputs and 50 outputs (i.e. $y = [y_1,y_2,...,y_{50}]$ and $\sigma = [\sigma_1,\sigma_2,...,\sigma_{50}]$) from the deep neural network, so an approach that is not very computationally expensive would be preferred. For practical matters, $y$ here is simply a temperature measurement of a gas in a box at 50 different locations in the box and $\sigma$ is the error of the temperature readings I get from my diagnostic; ideally I want to predict $y$ based on numerous data points from my diagnostic such that given enough data my predicted $\sigma$ can be minimized.)

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