Well known to the machine learning community, the random feature model is a
parametric approximation to kernel interpolation or regression methods. It is
typically used to approximate functions mapping a finite-dimensional input
space to the real line. In this paper, we instead propose a methodology for use
of the random feature model as a data-driven surrogate for operators that map
an input Banach space to an output Banach space. Although the methodology is
quite general, we consider operators defined by partial differential equations
(PDEs); here, the inputs and outputs are themselves functions, with the input
parameters being functions required to specify the problem, such as initial
data or coefficients, and the outputs being solutions of the problem. Upon
discretization, the model inherits several desirable attributes from this
infinite-dimensional viewpoint, including mesh-invariant approximation error
with respect to the true PDE solution map and the capability to be trained at
one mesh resolution and then deployed at different mesh resolutions. We view
the random feature model as a non-intrusive data-driven emulator, provide a
mathematical framework for its interpretation, and demonstrate its ability to
efficiently and accurately approximate the nonlinear parameter-to-solution maps
of two prototypical PDEs arising in physical science and engineering
applications: viscous Burgers' equation and a variable coefficient elliptic
equation.

本文提出了一种利用随机特征模型作为映射输入 Banach 空间到输出 Banach 空间之间算子的数据驱动代用模型的方法，将其视为非侵入式的数据驱动模拟器，提供了其解释的数学框架，并演示了其有效和准确地近似两个物理科学和工程应用中出现的原型偏微分方程非线性参数到解映射的能力：粘性 Burgers 方程和变系数椭圆方程。