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.

翻译：算子学习通过将推理摊销到无限问题族中，正在重塑科学计算领域。尽管神经算子（NOs）在回归任务中的理解日益深入，但对其在分类任务及其无监督对应任务——聚类——中的认识却远为不足。我们证明，在温和的核采样假设下，基于样本的神经算子能够在无限维再生核希尔伯特空间中学习任意有限类别的集合，即使这些类别既非凸也非连通。我们的通用聚类定理表明，在闭集的上Kuratowski拓扑中，任意$K$个闭类均可由NO参数化的类以任意精度逼近，这一概念可解释为不允许假阳性误分类。在此基础上，我们开发了一种面向函数数据的NO驱动聚类流程，并将其应用于未标记的常微分方程（ODE）轨迹族。离散化轨迹通过固定的预训练编码器提升为连续特征映射，并由轻量级可训练头部映射为软分配。在多种合成ODE基准测试上的实验表明，所提出的实用SNO能够在经典方法失效的机制中恢复潜在的动力学结构，这为我们的通用聚类理论提供了支持性证据。