Higher-order interactions underlie complex phenomena in systems such as biological and artificial neural networks, but their study is challenging due to the lack of tractable standard models. By leveraging the maximum entropy principle in curved statistical manifolds, here we introduce curved neural networks as a class of prototypical models for studying higher-order phenomena. Through exact mean-field descriptions, we show that these curved neural networks implement a self-regulating annealing process that can accelerate memory retrieval, leading to explosive order-disorder phase transitions with multi-stability and hysteresis effects. Moreover, by analytically exploring their memory capacity using the replica trick near ferromagnetic and spin-glass phase boundaries, we demonstrate that these networks enhance memory capacity over the classical associative-memory networks. Overall, the proposed framework provides parsimonious models amenable to analytical study, revealing novel higher-order phenomena in complex network systems.

翻译：高阶相互作用是生物与人工神经网络等系统中复杂现象的基础，但由于缺乏可处理的标准化模型，其研究面临挑战。通过利用弯曲统计流形中的最大熵原理，本文引入弯曲神经网络作为研究高阶现象的一类原型模型。通过精确的平均场描述，我们证明这些弯曲神经网络实现了一种自调节退火过程，能够加速记忆检索，从而引发具有多稳态和滞后效应的爆炸性有序-无序相变。此外，通过使用复本技巧在铁磁相与自旋玻璃相边界附近解析探索其记忆容量，我们证明这些网络相较于经典联想记忆网络具有更高的记忆容量。总体而言，所提出的框架提供了适合解析研究的简约模型，揭示了复杂网络系统中新颖的高阶现象。