Overall learning framework

The pieces work together as follows:

• An Extended Kalman Filter (EKF) operating on the higher order structure, tracks the latent state recursively as observations stream in. Each new measurement updates the estimate, taking into account both the topological dynamics and the observation noise.
• An online Expectation–Maximization loop learns the model's unknown parameters on the fly: how much uncertainty lives at each cell, what the convolution filter looks like, and the shape of the nonlinearity.
• The nonlinearity itself is parameterized using Random Fourier Features, giving a fixed-size, lightweight approximation that can keep up with streaming data.

The whole system is trained without ever seeing the latent states directly. It learns what it cannot observe from the underlying topological structure and the partial observations.                                         .

Discovering the hidden topology

Online 2-cell identification algorithm. Following a T_s-step warm-up, candidate 2-cells are evaluated across N_2 discrete windows. In each window, the EKF state is first cached and a candidate 2-cell is activated. If the forecasting NMSE_p reduction meets the threshold the 2-cell is accepted. If rejected, the system executes a rollback to the cached state and re-runs the EKF using the original topology present at the start of the window.

Perhaps, the most fascinating challenge is what to do when the underlying shape of the network is incomplete. In many critical infrastructure networks, only lower-order topological structures—like nodes and edges—are actually known.

To solve this, we introduce a heuristic cell identification algorithm. This mechanism acts as a topological detective: it uses the learned process uncertainty as a proxy for missing higher-order constraints. If the topology lacks a required higher-order cell, the filter inflates localized edge uncertainties. The algorithm then sequentially activates candidate 2-cells (faces) and retains them if they successfully reduce the forecasting error, explicitly inferring the second-order cell structures directly from the data stream.

Does it work?

Validations on synthetic and real datasets from water, sensor, and transportation networks demonstrate that this approach yields reliable estimates under partial observability and successfully recovers the underlying topological structures. It consistently achieves the best forecasting performance because it explicitly models the underlying system dynamics, outperforming standard linear and purely graph-based models.

Forecasting comparison between complete and incomplete signals on a Wireless Sensor Network dataset. The top two subplots display the observation tracking performance and Normalized Mean Square Error (NMSE) with complete signal, while the bottom two subplots illustrate the same metrics for an incomplete signal containing 30% missing values. In the missing data case the observation tracking curves plotted is for an unobserved sensor. A notable feature in both scenarios is the appearance of periodic peaks in the forecasting error, which correspond to underlying distributional shifts in the data. Following each peak, the NMSE gradually converges to a lower value, demonstrating the model's ability to adapt to transient changes in the data distribution. Although this convergence is slower in the presence of missing data, the model successfully continues to track the true observations, showcasing its strong robustness.

Why this matters

Critical systems—epidemics, neural networks, power grids, transportation systems—are increasingly instrumented but still deeply under-sensed. We have a few measurements (confirmed cases, recorded neurons, grid sensors, traffic cameras), a lot of structure, and a lot of dynamics we need to understand in real time: to detect outbreaks early, predict seizures, anticipate blackouts, manage flow.

A topology-aware state-space model offers a principled way to combine all three: streaming data, physical structure, and uncertainty. It produces estimates not just of what we see, but of what we don't—and it can refine its picture of the network itself as new data arrives.

Open challenges

The framework addresses a real gap in state estimation: systems where observations are noisy and the underlying structure matters. In principle, this applies wherever network topology constrains signal flow—from infrastructure (water, power) to biology (neural circuits) to epidemiology (transmission). By treating the network's shape as a built-in constraint rather than an afterthought, we can track dynamics that graph-based methods miss. Yet the path to transformative impact opens several research frontiers: 

1. Continual learning on evolving networks. Extending online learning to detect and adapt when the topology itself changes—new circuits form, transmission routes emerge, infrastructure rewires. Scaling sparse inference. 
2. Joint topology and dynamics discovery. Building systems that simultaneously infer hidden structure and learn latent dynamics from streaming observations. 

At SURE-AI, we are committed to advancing these methods precisely because they address this gap: principled, data-efficient inference for systems where trust and interpretability are non-negotiable. Our focus is on algorithms that are not just theoretically sound, but tractable and deployable on the infrastructure that keeps cities running, hospitals operating, and critical systems resilient.

It's another step toward seeing signals not just as data in time or space, but as data with shape—and with dynamics that move along that shape.