  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.
