Kernel Learning of PDE Solution Operators

TL;DR

This paper introduces a kernel-based method for learning PDE solution operators that incorporates physical priors, enabling extrapolation beyond observed data and providing theoretical error guarantees.

Contribution

It develops a regularization-based kernel approach that constructs an operator learner independent of input functions, advancing PDE solution methods.

Findings

Achieves high accuracy in Darcy flow and Helmholtz equations.

Provides convergence rates with suitable regularization.

Outperforms existing operator learning methods in accuracy and efficiency.

Abstract

A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a kernel ridge regression framework, and employs a regularization-based formulation to construct an operator learner. This yields a closed-form estimator that is independent of the input functions that determine the underlying PDE. From the perspective of regularization theory, the resulting estimator induces a well-defined operator that links input and output spaces, which contain the functions that define a Dirichlet problem and its solution, respectively. Consequently, it effectively shifts from a PDE solver to an operator-based solver. In contrast to standard supervised learning methods, it does not rely on paired input--output training data and enables systematic extrapolation beyond observed regimes. A full error analysis is conducted, providing convergence rates for the operator-based solver under suitable choices of regularization parameters. Extensive numerical experiments, including Darcy flow and Helmholtz equations, demonstrate that the proposed method achieves high accuracy and efficiency across a range of problem settings, and compares favorably with operator learning approaches in both approximation quality and computational cost.