In the pursuit of accurate experimental and computational data while minimizing effort, there is a constant need for high-fidelity results. However, achieving such results often requires significant computational resources. To address this challenge, this paper proposes a deep operator learning-based framework that requires a limited high-fidelity dataset for training. We introduce a novel physics-guided, bi-fidelity, Fourier-featured Deep Operator Network (DeepONet) framework that effectively combines low and high-fidelity datasets, leveraging the strengths of each. In our methodology, we began by designing a physics-guided Fourier-featured DeepONet, drawing inspiration from the intrinsic physical behavior of the target solution. Subsequently, we train this network to primarily learn the low-fidelity solution, utilizing an extensive dataset. This process ensures a comprehensive grasp of the foundational solution patterns. Following this foundational learning, the low-fidelity deep operator network's output is enhanced using a physics-guided Fourier-featured residual deep operator network. This network refines the initial low-fidelity output, achieving the high-fidelity solution by employing a small high-fidelity dataset for training. Notably, in our framework, we employ the Fourier feature network as the Trunk network for the DeepONets, given its proficiency in capturing and learning the oscillatory nature of the target solution with high precision. We validate our approach using a well-known 2D benchmark cylinder problem, which aims to predict the time trajectories of lift and drag coefficients. The results highlight that the physics-guided Fourier-featured deep operator network, serving as a foundational building block of our framework, possesses superior predictive capability for the lift and drag coefficients compared to its data-driven counterparts.

翻译：在追求精确实验与计算数据并最小化工作量的过程中，始终需要高保真度的结果。然而，获取此类结果往往需要大量计算资源。针对这一挑战，本文提出一种基于深度算子学习的框架，仅需有限的保真度数据集进行训练。我们引入了一种新颖的物理引导、双保真度、傅里叶特征深度算子网络（DeepONet）框架，该框架有效结合了低保真度与高保真度数据集，并充分利用各自的优势。在方法论中，我们首先借鉴目标解的内在物理行为，设计了一种物理引导的傅里叶特征DeepONet。随后，利用大规模数据集训练该网络，使其主要学习低保真度解，从而确保对基础解模式的全面掌握。在基础学习之后，通过物理引导的傅里叶特征残差深度算子网络增强低保真度深度算子网络的输出。该网络通过使用少量高保真度数据集进行训练，对初始低保真度输出进行精炼，最终获得高保真度解。值得注意的是，在我们的框架中，傅里叶特征网络被用作DeepONet的主干网络（Trunk网络），因其能够高精度地捕捉并学习目标解的振荡特性。我们通过一个经典的二维基准圆柱问题验证了所提方法，该问题旨在预测升力与阻力系数的时间轨迹。结果表明，作为框架基础模块的物理引导傅里叶特征深度算子网络，在升力与阻力系数的预测能力上优于其纯数据驱动的对应方法。