做科学计算和 PDE 求解的团队，终于有了一个能自然处理复杂几何和拓扑结构的算子学习框架，比传统 NO 更准，值得一试。
研究人员提出拓扑神经算子（TNOs），一种在细胞复形上进行算子学习的框架。TNOs 将数据表示为不同维度细胞上的特征，通过离散外微积分建模其相互作用，支持梯度、旋度和散度类型的跨维度耦合。其核心设计是将信息流动（由固定拓扑算子控制）与信息变换（可学习）解耦，从而尊重物理量的几何支撑并暴露守恒与兼容结构。进一步提出的层次化 TNOs（HTNOs）通过学习的粗化复形传播长程和拓扑依赖信息。在多个 PDE 基准测试（包括不规则几何流问题）中，TNOs 和 HTNOs 提升了精度，并证明了原生高阶和拓扑结构的优势。
Topological Neural Operators
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