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.

翻译：神经算子的研究为开发相较于传统方法更高效的偏微分方程求解途径奠定了基础。然而，现有的大多数神经算子缺乏为其预测结果提供不确定性度量的能力，这一能力在数据驱动场景（尤其是可用数据有限的场景）中至关重要。本文提出了一种新型的神经算子诱导高斯过程（NOGaP），该方法在利用算子学习强大学习能力的同时，充分挖掘了高斯过程的概率特性。所提出的框架不仅能提升预测精度，还可提供可量化的不确定性度量。通过在多个偏微分方程实例（包括伯格斯方程、达西流、非齐次泊松方程和波-平流方程）上的实验，我们对所提框架进行了全面评估。此外，通过与最先进的算子学习算法进行对比研究，进一步凸显了NOGaP的优势。实验结果展示了其卓越的精度与理想的不确定性特征，表明该框架具有广阔的应用前景。