Nonlinear PDEs approach to statistical mechanics of dense associative memories

Dense associative memories (DAMs) are widely used models in artificial intelligence for pattern recognition tasks; computationally, they have been proven to be robust against adversarial inputs 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 partial differential equations, to investigate their functioning. In particular, we prove differential identities involving DAM's 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.