发表机构
Nanjing University of Aeronautics and Astronautics; Nanjing Tech University; The Hong Kong Polytechnic University; KTH Royal Institute of Technology（南京航空航天大学; 南京工业大学; 香港理工大学; 瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对跨领域PDE学习中的几何-物理混杂问题，提出显式几何到算子变换的去混杂框架，提升预测精度、数据效率并大幅降低残差。
AI 中文摘要
跨不同领域学习偏微分方程（PDE）动力学对于预测建模和控制方程的数据驱动发现至关重要。然而，几何变化同时改变了场的表示和控制微分算子，将几何效应与观测动力学中的内在物理属性混杂在一起。本研究将几何-物理混杂确定为跨不同领域PDE学习的一个统一失败机制。在前向算子学习中，这种混杂增加了从有限数据推断几何相关算子变化的负担，降低了数据效率和泛化能力。在方程发现中，忽略几何诱导算子会错误指定候选库，导致参数偏差、遗漏控制项和虚假项。我们提出一个去混杂框架，使已知的几何到算子变换显式化。几何诱导系数场在五个算子学习基准上提高了预测和数据效率，而几何完整的候选库在演化域系统中恢复了生成方程，并将保留PDE残差降低了两个数量级以上。通过将已知几何作用与内在物理分离，所提出的框架支持在具有变化几何的科学和工程问题中实现更可靠和数据高效的PDE学习。
英文摘要
Learning partial differential equation (PDE) dynamics across varying domains is central to predictive modelling and data-driven discovery of governing equations. However, geometric variation alters both field representation and the governing differential operators, confounding geometric effects with intrinsic physical properties in the observed dynamics. This work identifies geometry-physics confounding as a unified failure mechanism for PDE learning across varying domains. In forward operator learning, this confounding increases the burden of inferring geometry-dependent operator changes from finite data, reducing data efficiency and generalisation. In equation discovery, omitting geometry-induced operators misspecifies the candidate library, leading to biased parameters, missed governing terms and spurious terms. We propose a de-confounding framework that makes the known geometry-to-operator transformation explicit. Geometry-induced coefficient fields improve prediction and data efficiency across five operator-learning benchmarks, while geometry-complete candidate libraries recover the generating equations and reduce held-out PDE residuals by more than two orders of magnitude in both evolving-domain systems. By separating known geometric action from intrinsic physics, the proposed framework supports more reliable and data-efficient PDE learning across scientific and engineering problems with varying geometries.
