The Random Feature Model for Input-Output Maps between Banach Spaces

TL;DR

This paper extends the random feature model to approximate operators between Banach spaces, especially PDE solution maps, offering mesh-invariant accuracy and multi-resolution deployment, demonstrated on two PDE examples.

Contribution

It introduces a novel methodology for using random feature models as surrogates for infinite-dimensional operators, with theoretical and practical advantages for PDEs.

Findings

Mesh-invariant approximation error achieved

Model trained at one mesh resolution generalizes to others

Efficiently approximates nonlinear PDE solution maps

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 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.