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

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