# Neural Operators Can Discover Functional Clusters

arXiv: https://arxiv.org/abs/2602.23528 · HTML: https://mlsanity.com/paper/2602.23528/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 21 of 21)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Clustering for Unsupervised Learning of Visual Features](https://mlsanity.com/paper/1807.05520/) (2018)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Invariant Information Clustering for Unsupervised Image Classification and Segmentation](https://mlsanity.com/paper/1807.06653/) (2018)
- [VisionTS: Visual Masked Autoencoders Are Free-Lunch Zero-Shot Time Series Forecasters](https://mlsanity.com/paper/2408.17253/) (2024)
- [A Comprehensive Survey on Deep Clustering: Taxonomy, Challenges, and Future Directions](https://mlsanity.com/paper/2206.07579/) (2022)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Leveraging Vision-Language Models for Granular Market Change Prediction](https://mlsanity.com/paper/2301.10166/) (2023)
- [Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models](https://mlsanity.com/paper/2306.05272/) (2023)
- [Functional Data Analysis of Amplitude and Phase Variation](https://mlsanity.com/paper/1512.03216/) (2015)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Review of Clustering Methods for Functional Data](https://mlsanity.com/paper/2210.00847/) (2022)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)
- [Shape Analysis of Functional Data with Elastic Partial Matching](https://mlsanity.com/paper/2105.08604/) (2021)

## Abstract (arXiv)

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.