Sorta. There are ways to extract kinds of uncertainty and confidence from NNs: for example, Gal's dropout trick where you train with dropout and then at runtime you use an 'ensemble' of multiple dropout-ed versions of your model, and the set of predictions gives a quasi-Bayesian posterior distribution for the predictions. NNs can be trained directly via HMC for small NNs, and there are arguments that constant-learning-rate SGD 'really' implements Bayesian inference and an ensemble of checkpoints yields an approximation of the posterior, etc. You can also train RL NNs which have an action of shortcutting computation and kicking the problem out to an oracle in exchange for a penalty, which trains them to specialize and 'know what they don't know' so they choose to call the oracle when they're insufficiently sure (this can be done for computational savings if the main NN is a small fast simple one and the oracle is a much bigger slower NN, or for safety if you imagine the oracle is a human or some fallback mechanism like halting).
I have some cites on these sorts of things in https://www.gwern.net/Tool-AI and you could also look at the relevant tags https://www.reddit.com/r/reinforcementlearning/search?q=flai... and https://www.reddit.com/r/reinforcementlearning/search?q=flai...
In particular, it's possible to learn the variance of the return using TD-methods with the same computational complexity as learning the expected value (the value function). See [0] for how to do it via the squared TD-error, or [1] for how to estimate it via the second moment of the return, and my own notes (soon to be published and expanded for my thesis) here [2].
It turns out that identifying states with high variance is a great way of locating model error-- most of the real-world environments are fairly deterministic, so states with high variance tend to be "aliased" combinations of different states with wildly different outcomes. You can use this to improve your agent via either allocating more representation power to those states to differentiate between very similar ones, or have your agent account for variance when choosing its policy. For example, Tesla could identify when variance spikes in its model and trigger an alert to the user that they may need to take over.
Additionally, there's work by Bellemare [3] for estimating the distribution of the return, which allows for all sorts of statistical techniques for quantifying confidence, risk, or uncertainty.
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0. https://arxiv.org/abs/1801.08287
1. https://arxiv.org/abs/1607.00446
Older ideas are http://mlg.eng.cam.ac.uk/yarin/blog_2248.html or http://papers.nips.cc/paper/7219-simple-and-scalable-predict...
Basically Bayesian neural networks were able to model confidence but are not applicable in current real-world scenarios. Thus, lots of methods rely on approximating bayesian methods.