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Deep learning (DL) has had unprecedented success and is now enteringscientific computing with full force. However, current DL methods typicallysuffer from instability, even when universal approximation properties guaranteethe existence of stable neural networks (NNs). We address this paradox bydemonstrating basic well-conditioned problems in scientific computing where onecan prove the existence of NNs with great approximation qualities, however,there does not exist any algorithm, even randomised, that can train (orcompute) such a NN. For any positive integers $K > 2$ and $L$, there are caseswhere simultaneously: (a) no randomised training algorithm can compute a NNcorrect to $K$ digits with probability greater than $1/2$, (b) there exists adeterministic training algorithm that computes a NN with $K-1$ correct digits,but any such (even randomised) algorithm needs arbitrarily many training data,(c) there exists a deterministic training algorithm that computes a NN with$K-2$ correct digits using no more than $L$ training samples. These resultsimply a classification theory describing conditions under which (stable) NNswith a given accuracy can be computed by an algorithm. We begin this theory byestablishing sufficient conditions for the existence of algorithms that computestable NNs in inverse problems. We introduce Fast Iterative REstarted NETworks(FIRENETs), which we both prove and numerically verify are stable. Moreover, weprove that only $\mathcal{O}(|\log(\epsilon)|)$ layers are needed for an$\epsilon$-accurate solution to the inverse problem.

Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem