Operator Learning Using Random Features: A Tool for Scientific Computing

TL;DR

This paper introduces a novel function-valued random features method for supervised operator learning, enabling efficient, scalable, and theoretically grounded approximation of operators in scientific computing, especially for nonlinear PDE problems.

Contribution

It extends classical random features to operator learning, providing a convex, quadratic training framework with convergence guarantees and connections to kernel methods.

Findings

Method demonstrates scalability and transferability.

Provides convergence guarantees and error bounds.

Effective for nonlinear PDE operator approximation.

Abstract

Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method.