Inferring latent dynamics from multivariate time-series defined over topological cell complexes is crucial for capturing the complex, higher-order interactions inherent in real-world systems such as in water, sensor, and transportation networks. However, reconstructing these latent states is challenging because the signals are coupled across higher-order topologies, while high dimensionality, nonlinear observations, and unknown structures increase the difficulty. To address this, we propose a topology-aware state space framework derived from stochastic partial differential equations on cell complexes. State evolution follows heat-like topological diffusion, with perturbations propagating along boundary operators. Under partial observability, we model observations using a cell complex convolution of latent states coupled with a nonlinear mapping. We perform recursive state estimation via an Extended Kalman Filter, simultaneously learning model parameters and uncertainties through an online Expectation-Maximization algorithm. Finally, for scenarios where only lower-order topological structure is known, e.g., nodes and edges, as in critical infrastructure networks, we introduce a heuristic cell identification algorithm to explicitly infer the second-order cell structures. Validations on synthetic and real datasets from water, sensor and transportation networks demonstrate that our approach yields reliable estimates under partial observability and successfully recovers the underlying topological structures.

翻译：从定义在拓扑复形上的多元时间序列中推断潜在动力学，对于捕捉现实系统（如供水网络、传感器网络和交通网络）中固有的复杂高阶相互作用至关重要。然而，由于信号在高阶拓扑结构上耦合，加之高维度、非线性观测和未知结构增加了难度，重建这些潜在状态颇具挑战性。为此，我们提出了一种基于复形上随机偏微分方程的拓扑感知状态空间框架。状态演化遵循类热拓扑扩散过程，扰动沿边界算子传播。在部分可观测条件下，我们利用潜在状态与非线性映射耦合的复形卷积来建模观测。通过扩展卡尔曼滤波进行递归状态估计，并借助在线期望最大化算法同时学习模型参数与不确定性。最后，针对仅已知低阶拓扑结构（如节点和边，常见于关键基础设施网络）的场景，我们引入了一种启发式胞腔识别算法，以显式推断二阶胞腔结构。在来自供水网络、传感器网络和交通网络的合成与真实数据集上的验证表明，该方法能在部分可观测条件下提供可靠估计，并成功恢复潜在的拓扑结构。