Non-linear PDEs approach to statistical mechanics of Dense Associative Memories

TL;DR

This paper introduces nonlinear PDE-based analytical methods to study Dense Associative Memories, enhancing understanding of their functioning and robustness by deriving differential identities related to their statistical mechanics.

Contribution

It develops a novel PDE approach to analyze DAMs, providing new differential identities and tools for their qualitative and quantitative study.

Findings

Derived differential identities involving DAM partition functions

Provided new insights into the mechanisms of DAMs

Enhanced interdisciplinary understanding of DAMs' behavior

Abstract

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.