Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map between function spaces. However, instead of full-field measurements, often the available data are only finite-dimensional parametrizations of model inputs or finite observables of model outputs. Building off of Fourier Neural Operators, this paper introduces the Fourier Neural Mappings (FNMs) framework that is able to accommodate such finite-dimensional inputs and outputs. The paper develops universal approximation theorems for the method. Moreover, in many applications the underlying parameter-to-observable (PtO) map is defined implicitly through an infinite-dimensional operator, such as the solution operator of a partial differential equation. A natural question is whether it is more data-efficient to learn the PtO map end-to-end or first learn the solution operator and subsequently compute the observable from the full-field solution. A theoretical analysis of Bayesian nonparametric regression of linear functionals, which is of independent interest, suggests that the end-to-end approach can actually have worse sample complexity. Extending beyond the theory, numerical results for the FNM approximation of three nonlinear PtO maps demonstrate the benefits of the operator learning perspective that this paper adopts.

翻译：参数化物理模型的计算高效替代方法在科学与工程中扮演着关键角色。算子学习提供了在函数空间之间建立映射的数据驱动替代方法。然而，实际可用数据往往并非全场测量值，而是模型输入的有限维参数化结果或模型输出的有限可观测值。本文基于傅里叶神经算子，提出能够适配此类有限维输入与输出的傅里叶神经映射框架，并证明了该方法的通用逼近定理。此外，在许多应用中，底层参数到可观测映射通过无限维算子（如偏微分方程的解算子）隐式定义。一个自然的问题是：直接端到端学习参数到可观测映射，与先学习解算子再基于全场解计算可观测值，哪种方式更具数据效率？本文对线性泛函贝叶斯非参数回归这一具有独立价值的问题进行了理论分析，结果表明端到端方法实际上可能具有更差的样本复杂度。在理论分析之外，针对三个非线性参数到可观测映射的傅里叶神经映射数值结果，进一步验证了本文采用的算子学习视角的优势。