I heard several times that one of the fundamental/open problems of deep learning is the lack of "general theory" on it, because, actually, we don't know why deep learning works so well. Even the Wikipedia page on deep learning has similar comments. Are such statements credible and representative of the state of the field?
There is a paper called Why does Deep Learning work so well?.
However, it is still not fully understood why deep learning works so well. In contrast to GOFAI (“good old-fashioned AI”) algorithms that are hand-crafted and fully understood analytically, many algorithms using artificial neural networks are understood only at a heuristic level, where we empirically know that certain training protocols employing large data sets will result in excellent performance. This is reminiscent of the situation with human brains: we know that if we train a child according to a certain curriculum, she will learn certain skills — but we lack a deep understanding of how her brain accomplishes this.
This is very much the case. Deep learning models even shallow ones such as stacked autoencoders and neural networks are not fully understood. There are efforts to understand what is happening to the optimization process for such a complex variable intensive function. But, this is a difficult task.
One way that researchers are using to discover how deep learning works is by using generative models. First we train a learning algorithm and handicap it systematically whilst asking it to generate examples. By observing the resulting generated examples we will be able to infer what is happening in the algorithm at a more significant level. This is very much like using inhibitors in neuroscience to understand what different components of the brain are used for. For example, we know that the visual cortex is where it is because if we damage it you will go blind.
It probably depends on what one means by "fundamental theory", but there is no lack of rigorous quantitative theory in deep learning, some of which is very general, despite claims to the contrary.
One good example is the work around energy-based methods for learning. See e.g. Neal & Hinton's work on variational inference and free energy: http://www.cs.toronto.edu/~fritz/absps/emk.pdf
Also this guide to energy minimization as a "common theoretical framework for many learning models" by Yann LeCun and colleagues: http://yann.lecun.com/exdb/publis/pdf/lecun-06.pdf
And a general framework for energy-based models by Scellier and Bengio: https://arxiv.org/pdf/1602.05179.pdf
There is also Hinton & Sejnowski's earlier work which shows analytically that a particular Hopfield-inspired network + unsupervised learning algorithm can approximate Bayes-optimal inference: https://papers.cnl.salk.edu/PDFs/Optimal%20Perceptual%20Inference%201983-646.pdf
There are many papers linking deep learning with theoretical neuroscience as well, such as the following, which shows that the effects of backpropagation can be achieved in biologically plausible neural architectures: https://arxiv.org/pdf/1411.0247.pdf
Of course there are many open questions and no single, uncontroverisal unified theory, but the same could be said of almost any field.
Your wikipedia quote is questionable because deep learning is well developed. In fact, there is a
 on the Wikipedia page.
Look at https://github.com/terryum/awesome-deep-learning-papers. There are like 100 papers in the link, do you still think deep-learning lacks "general theory"?
Yes. Deep learning is hard to understand because it is a very complicated model. But that doesn't mean we don't have the theories.
lime package and it's paper: "Why Should I Trust You?": Explaining the Predictions of Any Classifier will help you. The paper suggests we should be able to approximate a complicated model (includes deep learning) locally with a much simpler model.
A key question that remains in the theory of deep learning is why such huge models (with many more parameters than data points) don't overfit on the datasets we use.
Classical theory based on complexity measures does not explain the behaviour of practical neural networks. For instance estimates of VC dimension give vacuous generalisation bounds. As far as I know, the tightest (upper and lower) bounds on the VC dimension are given in  and are on the order of the number of weights in the network. Clearly this worst case complexity cannot explain how e.g. a big resnet generalises on CIFAR or MNIST.
Recently there have been other attempts at ensuring generalisation for neural networks, for instance by relation to the neural tangent kernel or by various norm measures on the weights. Respectively, these have been found to not apply to practically sized networks and to have other unsatisfactory properties .
There is some work in the PAC Bayes framework for non-vacuous bounds, e.g. . These setups, however, require some knowledge of the trained the network and so are different in flavour to the classical PAC analysis.
Some other aspects:
optimisation: how come we get 'good' solutions from gradient descent on such a non-convex problem? (There are some answers to this in recent literature)
interpretability: Can we explain on an intuitive level what the network is 'thinking'? (Not my area)
I'd like to point out there isn't a good theory on why machine learning works in general. VC bounds still assume a model, but reality doesn't fit any of these mathematical ideals. Ultimately when it comes to application everything comes down to emperical results. Even quantifying the similarity between images using an algorithm which is consistent with humans intuitive understanding is really hard
Anyway NN dont work well in their fully connected form. All successful networks have some kind of regularization built into the network architecture (CNN, LSTM, etc).