发表机构
Institute for Life and Medical Sciences, Kyoto University; Department of Mathematics, The University of Hong Kong; Bioinformatics Center, Institute for Chemical Research, Kyoto University（京都大学生命医科学研究所; 香港大学数学系; 京都大学化学研究所生物信息中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析了线性阈值与ReLU循环神经网络中，节点更新规则和状态域对全局有限时间可观测性所需最少观测节点数量的影响，推导了相关下界并构建了对应网络验证紧性。
AI 中文摘要
本文研究了节点更新规则和容许状态域如何影响带线性阈值和ReLU更新函数的循环神经网络实现全局有限时间可观测性所需的最少观测节点数量。在常见的二值状态域上，我们构建了一类K线性阈值（K-LT）网络，其初始状态可通过单个观测节点的有限输出轨迹唯一重构。我们进一步建立了二值K-ReLU网络与K-AND布尔网络之间的动力学等价关系，将现有的类别级观测节点下界推广到二值K-ReLU网络。对于非负取值的ReLU网络，当相关预激活保持非负时，可观测性问题可归约为经典线性系统场景。对于一般实值ReLU网络，我们证明当每个节点更新涉及的状态变量数量无限制时，全局有限时间可观测性至少需要n/2个观测节点；当n=2K时该下界是紧的，我们为此构建了一个恰好从K个节点可观测的K-ReLU网络。这些结果表明，更新规则和状态域从根本上影响着极端观测要求：时间演化可将有限状态信息集中到单个测量轨迹中，而激活诱导的秩损失则在连续状态ReLU网络中形成了固有传感器下界。
英文摘要
This paper investigates how node update rules and admissible state domains affect the minimum number of observation nodes required for global finite-horizon observability in recurrent neural networks with linear-threshold and ReLU update functions. Over a common binary state domain, we construct a class of $K$-linear-threshold ($K$-LT) networks whose initial states can be uniquely reconstructed from the finite output trajectory of a single observation node. We further establish a dynamical equivalence between binary-valued $K$-ReLU networks and $K$-AND Boolean networks, which transfers existing class-level observation-node bounds to binary-valued $K$-ReLU networks. For nonnegative-valued ReLU networks, the observability problem reduces to the classical linear-system setting whenever the relevant pre-activations remain nonnegative. For general real-valued ReLU networks, we prove that global finite-horizon observability requires at least $n/2$ observation nodes when no restriction is imposed on the number of state variables involved in each node update. This lower bound is tight when $n=2K$, for which we construct a $K$-ReLU network observable from exactly $K$ nodes. These results show that both update rules and state domains fundamentally affect extremal observation requirements: temporal evolution can concentrate finite-state information into a single measured trajectory, whereas activation-induced rank loss creates an intrinsic sensor lower bound in continuous-state ReLU networks.
