The Random Feature Model for Input-Output Maps between Banach Spaces
The Random Feature Model for Input-Output Maps between Banach Spaces
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DOI：
10.1137/20m133957x
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发表时间：
2020-05
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通讯作者：
Nicholas H. Nelsen;Andrew M. Stuart
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作者：
Nicholas H. Nelsen;Andrew M. Stuart
机器学习社区众所周知，随机特征模型最初由Rahimi和Recht在2008年引入，是对核插值或回归方法的参数近似。它通常用于将有限维输入空间映射到真实的直线的近似函数。在本文中，我们反而提出了一种方法，使用随机特征模型作为数据驱动的代理算子，将输入Banach空间映射到输出Banach空间。虽然该方法是相当普遍的，我们认为运营商定义的偏微分方程（PDE），在这里，输入和输出本身是函数，与输入参数的功能需要指定的问题，如初始数据或系数，和输出的解决方案的问题。离散化后，该模型继承了几个理想的属性，从这个无限维，功能空间的观点，包括网格不变的近似误差相对于真正的PDE解决方案的地图和能力，在一个网格分辨率进行训练，然后部署在不同的网格分辨率。我们认为随机特征模型作为一个非侵入性的数据驱动的仿真器，提供了一个数学框架，其解释，并证明其能力，有效地和准确地近似的非线性参数的解决方案映射的两个原型偏微分方程在物理科学和工程应用：粘性Burgers方程和变系数椭圆方程。
Well known to the machine learning community, the random feature model, originally introduced by Rahimi and Recht in 2008, 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, function space 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.