发表机构
Institute of Mathematics; Institute of Materials; École Polytechnique Fédérale de Lausanne（数学研究所; 材料研究所; 洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对傅里叶神经算子跨域迁移能力差的问题，提出域无关的欧几里得傅里叶神经算子，经热方程和材料科学任务验证，可泛化到未见过的网格尺寸与域。
AI 中文摘要
傅里叶神经算子（FNOs）提供了一种学习函数空间之间映射的高效框架，其结构本身与训练和评估时所用的网格分辨率无关。然而，FNOs并非与所应用的周期域无关：它们的离散谱权重由整数傅里叶模式数索引，对应于物理波矢。当应用于不同的域时，相同的训练权重作用于不同的波矢，FNO会隐式地表示一个不同的算子，这使得FNO不适合跨域迁移至关重要的任务。我们提出欧几里得傅里叶神经算子（EFNOs）作为FNO的域无关替代方案。通过将谱核参数化为物理波矢的连续函数，EFNO可以学习在不同形状和大小的周期域上一致作用的算子。我们在简单的热方程任务和实际相关的材料科学任务（学习不同晶体结构之间的交换关联势）上对EFNO进行评估，结果表明EFNO能够泛化到未见过的网格尺寸和域。
英文摘要
Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.
