Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations

TL;DR

This paper introduces NOGaP, a novel framework combining neural operators and Gaussian Processes to solve parametric PDEs with improved accuracy and uncertainty quantification, outperforming existing methods.

Contribution

The paper presents a new probabilistic operator learning framework that integrates Gaussian Processes with neural operators for better uncertainty estimation in PDE solutions.

Findings

NOGaP achieves higher accuracy than existing neural operator methods.

The framework provides reliable uncertainty quantification.

Experimental results on various PDEs demonstrate its effectiveness.

Abstract

The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.