The Random Feature Model for InputOutput Maps between Banach Spaces
Abstract
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 finitedimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a datadriven 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 infinitedimensional viewpoint, including meshinvariant 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 nonintrusive datadriven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parametertosolution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.10224
 Bibcode:
 2020arXiv200510224N
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Machine Learning;
 Physics  Computational Physics;
 Statistics  Machine Learning;
 65D15;
 65D40;
 62M45;
 35R60
 EPrint:
 To appear in SIAM Journal on Scientific Computing