Neural Operators Can Discover Functional Clusters

TL;DR

This paper demonstrates that neural operators can be theoretically and practically used to cluster functions in infinite-dimensional spaces, enabling effective unsupervised classification of complex functional data.

Contribution

It provides a universal clustering theorem for neural operators in infinite-dimensional spaces and develops a practical clustering pipeline for functional data, especially ODE trajectories.

Findings

Neural operators can learn multiple classes in infinite-dimensional RKHS.

The universal clustering theorem disallows false-positive misclassifications.

The proposed clustering method successfully recovers latent structures in ODE data.

Abstract

Operator learning is reshaping scientific computing by amortizing inference across infinite families of problems. While neural operators (NOs) are increasingly well understood for regression, far less is known for classification and its unsupervised analogue: clustering. We prove that sample-based neural operators can learn any finite collection of classes in an infinite-dimensional reproducing kernel Hilbert space, even when the classes are neither convex nor connected, under mild kernel sampling assumptions. Our universal clustering theorem shows that any $K$ closed classes can be approximated to arbitrary precision by NO-parameterized classes in the upper Kuratowski topology on closed sets, a notion that can be interpreted as disallowing false-positive misclassifications. Building on this, we develop an NO-powered clustering pipeline for functional data and apply it to unlabeled families of ordinary differential equation (ODE) trajectories. Discretized trajectories are lifted by a fixed pre-trained encoder into a continuous feature map and mapped to soft assignments by a lightweight trainable head. Experiments on diverse synthetic ODE benchmarks show that the resulting practical SNO recovers latent dynamical structure in regimes where classical methods fail, providing evidence consistent with our universal clustering theory.