Neural Operators Can Discover Functional Clusters

2026-02-26

AI Summary

This work addresses the challenge of unsupervised clustering in infinite-dimensional function spaces under non-convex and disconnected cluster structures. We propose a general neural operator–based clustering framework that maps discrete trajectories into continuous features in a reproducing kernel Hilbert space via a pretrained encoder, followed by a lightweight trainable head to produce soft cluster assignments. Theoretically, we establish, for the first time, that neural operators can approximate any finite collection of closed sets in this space to arbitrary precision, thereby formulating a universal clustering theory in the upper Kuratowski topology. Experimentally, our method successfully recovers latent dynamical structures across multiple synthetic ordinary differential equation trajectory benchmarks, significantly outperforming classical approaches and demonstrating both effectiveness and practical utility.

Technology Category

• Machine Learning: Clustering
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• neural operators
• functional clustering
• unsupervised learning
• infinite-dimensional data
• ODE trajectories

Innovation

• neural operators
• functional clustering
• universal approximation
• Kuratowski topology
• ODE trajectories