We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

翻译：我们提出拓扑神经算子（TNOs），一种基于胞腔复形的算子学习原则性框架，将神经算子从点/边上的函数提升至拓扑域。TNOs将数据表示为定义在不同维度胞腔上的特征，并通过离散外微分计算建模其相互作用，利用梯度型、旋度型及散度型算子实现显式的跨维耦合。其核心设计原则是解耦信息流动路径（由固定拓扑算子控制）与信息变换方式（可通过学习获得），从而生成尊重物理量几何支撑并揭示守恒与相容性结构的模型。我们进一步提出层次化拓扑神经算子（HTNOs），通过引入可学习的粗化复形来传播长程及拓扑依赖信息。本框架将现有神经算子作为特例纳入其中，为跨离散化方式的算子学习提供统一视角。在包括非规则几何流动问题在内的多类偏微分方程基准测试中，TNOs与HTNOs显著提升精度；控制实验进一步验证了原生高阶及拓扑结构的优势。项目主页：https://circle-group.github.io/research/TNO