  Dense associative memories (DAM), are widespread models in artificial
intelligence used for pattern recognition tasks; computationally, they have
been proven to be robust against adversarial input and theoretically,
leveraging their analogy with spin-glass systems, they are usually treated by
means of statistical-mechanics tools. Here we develop analytical methods, based
on nonlinear PDEs, to investigate their functioning. In particular, we prove
differential identities involving DAM partition function and macroscopic
observables useful for a qualitative and quantitative analysis of the system.
These results allow for a deeper comprehension of the mechanisms underlying
DAMs and provide interdisciplinary tools for their study.
