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 NOGaP.

中文翻译：

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用于参数偏微分方程概率解的神经算子诱导高斯过程框架


与传统方法相比，神经算子的研究为解决偏微分方程（PDE）的有效方法的开发铺平了道路。然而，大多数现有的神经算子缺乏为其预测提供不确定性度量的能力，这是一个至关重要的方面，特别是在可用数据有限的数据驱动场景中。在这项工作中，我们提出了一种新颖的神经算子诱导高斯过程（NOGaP），它利用高斯过程（GP）的概率特征，同时利用算子学习的学习能力。所提出的框架提高了预测准确性，并提供了可量化的不确定性度量。通过对各种偏微分方程示例的实验对所提出的框架进行了广泛的评估，包括伯格方程、达西流、非齐次泊松方程和波平流方程。此外，还与最先进的算子学习算法进行了比较研究，以突出 NOGaP 的优势。结果证明了卓越的准确性和预期的不确定性特征，表明所提出的框架 NOGaP 具有广阔的前景。