In this work we develop analytical techniques to investigate a broad class of associative neural networks set in the high-storage regime. These techniques translate the original statistical–mechanical problem into an analytical–mechanical one which implies solving a set of partial differential equations, rather than tackling the canonical probabilistic route. We test the method on the classical Hopfield model – where the cost function includes only two-body interactions (i.e., quadratic terms) – and on the “relativistic” Hopfield model — where the (expansion of the) cost function includes $p$-body (i.e., of degree $p$) contributions. Under the replica symmetric assumption, we paint the phase diagrams of these models by obtaining the explicit expression of their free energy as a function of the model parameters (i.e., noise level and memory storage). Further, since for non-pairwise models ergodicity breaking is non necessarily a critical phenomenon, we develop a fluctuation analysis and find that criticality is preserved in the relativistic model.

在这项工作中，我们开发了分析技术，以研究在高存储状态下设置的各种关联神经网络。这些技术将原始的统计-机械问题转化为分析-机械问题，这意味着要解决一组偏微分方程，而不是解决规范的概率路线。我们在经典的Hopfield模型（成本函数仅包括两体相互作用（即二次项））和“相对论” Hopfield模型（其中的（扩展）成本函数包括）上测试了该方法。$p$-身体（即程度 $p$）捐款。在复制对称假设下，我们通过获得它们的自由能作为模型参数（即噪声水平和内存存储）的函数的显式表达，来绘制这些模型的相图。此外，由于对于非成对模型，遍历遍历不一定是一种临界现象，因此我们进行了波动分析，发现在相对论模型中保留了临界。