Neural Operators Can Discover Functional Clusters - Yicen Li and Ruiyang Hong

Описание: Clustering functional data generated by dynamical systems poses a fundamental mathematical challenge: trajectories live in infinite-dimensional spaces where classical Euclidean notions of similarity fail to capture shape equivalence. Despite its importance, clustering families of ODE solutions remains largely unexplored from a theoretical perspective, and existing functional data analysis pipelines typically treat trajectories as generic signals, offering no guarantees that learned partitions converge to the true clustering structure of the underlying dynamics.
In this work, we propose a theoretically grounded framework for universal clustering of ODE trajectories using Sampling-Based Neural Operators (SNOs). We address the inherently set-valued nature of clustering by proving that, under sufficient expressivity, the decision regions induced by SNOs converge to the true cluster partitions in the upper Kuratowski topology, a notion of convergence that controls the approximation of cluster regions themselves rather than pointwise function values, and is therefore well suited to infinite-dimensional settings.
We further demonstrate that this abstract operator framework admits a practical and scalable instantiation: trajectories are discretized onto finite grids and processed through a fixed, pre-trained visual encoder acting as a continuous feature map, followed by a lightweight clustering head. Experiments on diverse synthetic ODE benchmarks show that the proposed operator-learning approach not only aligns with the predicted convergence behaviour under refined sampling, but also effectively captures latent dynamical structure in regimes where classical methods fail.

Joint work: J. Antonio Lara Benitez, Anastasis Kratsios, Paul David McNicholas, and Maarten de Hoop