标题： 融合深度ONet：一种用于任意网格上几何相关高超音速流的数据高效神经算子 显示英文标题

标题： Fusion DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic Flows on Arbitrary Grids

摘要： 设计再入飞行器需要准确预测其几何结构周围的高超音速流动。 快速预测此类流动可以彻底改变飞行器设计，特别是对于可变形几何结构。 我们评估了先进的神经算子模型，如深度算子网络（DeepONet）、参数条件U-Net、傅里叶神经算子（FNO）和MeshGraphNet，目的是解决在有限数据下学习与几何相关的高超音速流动场的挑战。 具体而言，我们比较了这些模型在两种网格类型上的性能：均匀笛卡尔网格和不规则网格。 为了训练这些模型，我们使用了36种独特的椭圆几何结构，利用高阶熵稳定DGSEM求解器生成高保真度模拟，强调在数据稀缺情况下的挑战。 我们评估并比较了这四种基于算子的模型在预测椭圆体周围高超音速流动场方面的有效性。 此外，我们开发了一种新的框架，称为融合DeepONet，它利用神经场概念并在不同几何结构上表现出有效的泛化能力。 尽管训练数据稀缺，融合DeepONet在均匀网格上的表现与参数条件U-Net相当，而在不规则、任意网格上则优于MeshGraphNet和原始DeepONet。 与U-Net、MeshGraphNet和FNO相比，融合DeepONet所需的可训练参数显著减少，使其计算效率更高。 我们还使用奇异值分解分析了融合DeepONet模型的基函数。 该分析表明，融合DeepONet能够有效泛化到未见过的解决方案，并适应不同的几何结构和网格点，在训练数据有限的情况下展现出其鲁棒性。 显示英文摘要

摘要： Designing re-entry vehicles requires accurate predictions of hypersonic flow around their geometry. Rapid prediction of such flows can revolutionize vehicle design, particularly for morphing geometries. We evaluate advanced neural operator models such as Deep Operator Networks (DeepONet), parameter-conditioned U-Net, Fourier Neural Operator (FNO), and MeshGraphNet, with the objective of addressing the challenge of learning geometry-dependent hypersonic flow fields with limited data. Specifically, we compare the performance of these models for two grid types: uniform Cartesian and irregular grids. To train these models, we use 36 unique elliptic geometries for generating high-fidelity simulations with a high-order entropy-stable DGSEM solver, emphasizing the challenge of working with a scarce dataset. We evaluate and compare the four operator-based models for their efficacy in predicting hypersonic flow field around the elliptic body. Moreover, we develop a novel framework, called Fusion DeepONet, which leverages neural field concepts and generalizes effectively across varying geometries. Despite the scarcity of training data, Fusion DeepONet achieves performance comparable to parameter-conditioned U-Net on uniform grids while it outperforms MeshGraphNet and vanilla DeepONet on irregular, arbitrary grids. Fusion DeepONet requires significantly fewer trainable parameters as compared to U-Net, MeshGraphNet, and FNO, making it computationally efficient. We also analyze the basis functions of the Fusion DeepONet model using Singular Value Decomposition. This analysis reveals that Fusion DeepONet generalizes effectively to unseen solutions and adapts to varying geometries and grid points, demonstrating its robustness in scenarios with limited training data.