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.

通过使用有限的高保真数据集进行训练，我们提出了一种基于深度算子学习的框架，它能够在减少工作量的同时获得准确的实验和计算数据。我们的方法首先使用物理引导的Fourier特征深度算子网络通过广泛的数据集学习低保真解决方案的基本模式，然后通过物理引导的Fourier特征残差深度算子网络提高低保真网络输出，使用少量的高保真数据集进行训练，从而获得高保真解决方案。结果表明，我们提出的框架中物理引导的Fourier特征深度算子网络相比于基于数据驱动的方法在预测升力和阻力系数方面具有更好的预测能力。

基于物理引导的双保真傅里叶特征算子学习框架用于预测阻力和升力系数的时间演化