Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics

TL;DR

This paper presents a novel Gaussian Process-based neural operator that achieves resolution independence and scalability for solving complex parametric PDEs in computational mechanics, with enhanced uncertainty quantification.

Contribution

It introduces a neural operator-embedded kernel with a stochastic dual descent training algorithm, addressing resolution dependence and cubic complexity of traditional GPs.

Findings

Outperforms standard GP models in accuracy and efficiency

Demonstrates resolution independence in high-dimensional PDEs

Provides robust uncertainty estimation in complex systems

Abstract

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty quantification. This paper introduces a novel Gaussian Process (GP) based neural operator for solving parametric differential equations. The approach proposed leverages the expressive capability of deterministic neural operators and the uncertainty awareness of conventional GP. In particular, we propose a ``neural operator-embedded kernel'' wherein the GP kernel is formulated in the latent space learned using a neural operator. Further, we exploit a stochastic dual descent (SDD) algorithm for simultaneously training the neural operator parameters and the GP hyperparameters. Our approach addresses the (a) resolution dependence and (b) cubic complexity of traditional GP models, allowing for input-resolution independence and scalability in high-dimensional and non-linear parametric systems, such as those encountered in computational mechanics. We apply our method to a range of non-linear parametric partial differential equations (PDEs) and demonstrate its superiority in both computational efficiency and accuracy compared to standard GP models and wavelet neural operators. Our experimental results highlight the efficacy of this framework in solving complex PDEs while maintaining robustness in uncertainty estimation, positioning it as a scalable and reliable operator-learning algorithm for computational mechanics.