Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate traditional numerical methods and to enable data-driven discovery. A popular variant of neural operators is the Fourier neural operator (FNO). Previous analysis proving universal operator approximation theorems for FNOs resorts to use of an unbounded number of Fourier modes and limits the basic form of the method to problems with periodic geometry. Prior work relies on intuition from traditional numerical methods, and interprets the FNO as a nonstandard and highly nonlinear spectral method. The present work challenges this point of view in two ways: (i) the work introduces a new broad class of operator approximators, termed nonlocal neural operators (NNOs), which allow for operator approximation between functions defined on arbitrary geometries, and includes the FNO as a special case; and (ii) analysis of the NNOs shows that, provided this architecture includes computation of a spatial average (corresponding to retaining only a single Fourier mode in the special case of the FNO) it benefits from universal approximation. It is demonstrated that this theoretical result unifies the analysis of a wide range of neural operator architectures. Furthermore, it sheds new light on the role of nonlocality, and its interaction with nonlinearity, thereby paving the way for a more systematic exploration of nonlocality, both through the development of new operator learning architectures and the analysis of existing and new architectures.

翻译：神经算子架构可逼近无限维巴拿赫函数空间之间的算子。由于其既能加速传统数值方法又能实现数据驱动发现的潜力，这类算子在计算科学与工程领域日益受到关注。傅里叶神经算子（FNO）是神经算子的一种流行变体。以往证明FNO通用算子逼近定理的分析方法，局限于使用无界数量的傅里叶模，并将该方法的基本形式限制于周期几何问题。先前工作依托传统数值方法的直觉，将FNO视为一种非标准的高度非线性谱方法。本研究从两方面挑战这一观点：（i）提出一类新型广义算子逼近器，称为非局部神经算子（NNO），该算子允许在任意几何定义域的函数之间进行算子逼近，并将FNO作为特例包含其中；（ii）对NNO的分析表明，当该架构包含空间平均计算（对应于FNO特例中仅保留单个傅里叶模式）时，其具备通用逼近能力。理论结果证明，这一发现统一了对广泛神经算子架构的分析。此外，该工作重新阐释了非局部性的作用及其与非线性性的交互机制，从而为通过发展新型算子学习架构与分析现有及新型架构来系统探索非局部性铺平了道路。