Lately, there have been quite a few theories on neural networks as ideal nonlinear approximators [1]. Similarly, people have shown many ways that gradient descent can tend to reach a global maximum of regularized curve-closeness[2]. Which is to say, if your development cycle is: gather-data, train, test, deploy, we know this approximates the data almost ideally; you can't really do much better than a deep network.
But we know in practice, when deployed, that deep neural networks actually have many limitations (compared to our intuitions or compared to human performance, etc). There are some obvious explanations. Of course, they're limited by our ability to gather data and by the biases of the data. But even more, they're limited to situations where you have large chunks of unchanging data that you can extrapolate from.
Given that deeps are more or less perfect for the train-test-deploy cycle, it seems like the problem is with this cycle itself. And it's easy to see human beings somehow acting "intelligently" without using this cycle. So figuring out an alternative to this might be something to look at.
[1] For example: Nonlinear Approximation and (Deep) ReLU Networks I. Daubechies, et al https://arxiv.org/abs/1905.02199
[2] For example: Gradient descent optimizes over-parameterized deep ReLU networks Difan Zou et al https://link.springer.com/article/10.1007/s10994-019-05839-6
e.g.
Robustness & expressiveness
Memorization & catastrophic forgetting
Ensembling & ntk & optimization
Double descent
Pruning
This is a great time and place to be! Neural networks are the twenty-first century Fourier series. It's just that we don't yet understand them. We can easily run them (synthesis) but we are missing the analysis. There's a lot of math to do here.
I'm in the exact same position.
I think mathematical explanations and (substantiated) intuition for the things that practitioners discover would be very useful. Maybe an all explaining grand theory of deep learning is possible.
It is? I didn't see that buzzword very much at NeurIPS this year.
The whole pipeline from Pytorch or TensorFlow and Python to LLVM, to GPUs or TPUs is absolutely rigorous. Much more rigorous, in fact, than normal, hand-written mathematics, as you find it in e.g. a typical Annals of Mathematics publication, or mathematical textbook!
I think what you really have in mind is a simple model of modern deep learning that is not fully accurate, but still useful!
Let me argue by analogy. You are looking for something that is to deep learning what the lambda-calculus is to the Haskell compiler. One of the main simplifications in programming language theory is replacing finite precision arithmetic (which is painfully complex) with mathematical integers and real numbers (which are much simpler). Would a theory of deep learning based on mathematical reals be valuable in a theory of deep learning? The stunning success of floating point formats like bfloat16 [1] suggests otherwise, since arithmetic precision in deep learning is closely connected to important learning phenomena such as overfitting and regularisation.
I am tempted to be provocative and say that you are really looking for less rigour!
[1] https://en.wikipedia.org/wiki/Bfloat16_floating-point_format
What you can do is use the existing rigour and derive stronger properties for restricted classes of NNs (just like you can prove stronger properties for simple subsystems of lambda-calculus) and that is an interesting field of study.
Non-recurrent neural networks are not Turing-complete in any sense.
In the sense that for non-trivial applications, we struggle to define what approach is successful and why it's successful without leaning on empirical metrics.
You cannot, in general, predict arbitrary program properties without running programs, that's the essence of Rice's theorem [1]. By the famous Universal Approximation Theorems for neural networks, in general, NNs are Turing complete, which precludes general and simple mathematical "silver bullets" that will help you overcome those challenges.