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 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.

中文翻译：

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面向算子学习的高斯过程：一种用于计算力学的不确定性感知分辨率独立算子学习算法


计算力学领域对准确、高效和可扩展的解决方案的需求不断增长，这凸显了对高级算子学习算法的需求，这些算法可以有效地处理大型数据集，同时提供可靠的不确定性量化。本文介绍了一种基于高斯过程 （GP） 的新型神经运算符，用于求解参数微分方程。所提出的方法利用了确定性神经算子的表达能力和传统 GP 的不确定性意识。特别是，我们提出了一种 “神经算子嵌入式内核”，其中 GP 内核是在使用神经算子学习的潜在空间中制定的。此外，我们利用随机双下降 （SDD） 算法同时训练神经算子参数和 GP 超参数。我们的方法解决了传统 GP 模型的 （a） 分辨率依赖性和 （b） 立方复杂性，允许在高维和非线性参数系统中实现输入分辨率独立性和可扩展性，例如在计算力学中遇到的系统。我们将我们的方法应用于一系列非线性参数偏微分方程 （PDE），并证明了与标准 GP 模型和小波神经运算符相比，它在计算效率和准确性方面的优势。我们的实验结果强调了该框架在求解复杂偏微分方程方面的有效性，同时保持了不确定性估计的稳健性，将其定位为计算力学的可扩展且可靠的运算符学习算法。