Discretization-independent multifidelity operator learning for partial differential equations

TL;DR

This paper introduces a discretization-independent operator learning model for PDEs that enhances multifidelity learning, providing theoretical guarantees and demonstrating improved accuracy and efficiency through extensive numerical experiments.

Contribution

The paper presents a novel discretization-independent operator learning framework that enables robust multifidelity PDE solutions with theoretical guarantees and empirical validation.

Findings

Multifidelity training improves accuracy and efficiency.

Discretization independence enhances robustness across PDE types.

Theoretical guarantees demonstrate universal approximation.

Abstract

We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. We introduce the concepts of \textit{numerical operator learning} and \textit{discretization independence}, which clarify the relationship between theoretical formulations and practical realizations of operator learning models. Our model is discretization-independent, making it particularly effective for multifidelity learning. We establish theoretical approximation guarantees, demonstrating uniform universal approximation under strong assumptions on the input functions and statistical approximation under weaker conditions. To our knowledge, this is the first comprehensive study that investigates how discretization independence enables robust and efficient multifidelity operator learning. We validate our method through extensive numerical experiments involving both local and nonlocal PDEs, including time-independent and time-dependent problems. The results show that multifidelity training significantly improves accuracy and computational efficiency. Moreover, multifidelity training further enhances empirical discretization independence.