Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve Riemann problems encountered in compressible flows for extreme pressure jumps (up to pressure ratio). In particular, we first consider the DeepONet that we train in a two-stage process, following the recent work of Lee and Shin (2023), wherein the first stage, a basis is extracted from the trunk net, which is orthonormalized and subsequently is used in the second stage in training the branch net. This simple modification of DeepONet has a profound effect on its accuracy, efficiency, and robustness and leads to very accurate solutions to Riemann problems compared to the vanilla version. It also enables us to interpret the results physically as the hierarchical data-driven produced basis reflects all the flow features that would otherwise be introduced using ad hoc feature expansion layers. We also compare the results with another neural operator based on the U-Net for low, intermediate, and very high-pressure ratios that are very accurate for Riemann problems, especially for large pressure ratios, due to their multiscale nature but computationally more expensive. Overall, our study demonstrates that simple neural network architectures, if properly pre-trained, can achieve very accurate solutions of Riemann problems for real-time forecasting. The source code, along with its corresponding data, can be found at the following URL: .

中文翻译：

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RiemannONets：黎曼问题的可解释神经算子

开发适当的表示方法来模拟具有强冲击波、稀疏性和接触不连续性的高速流动一直是数值分析中长期存在的问题。在这里，我们采用神经算子来解决极端压力跳跃（高达压力比）的可压缩流中遇到的黎曼问题。特别是，我们首先考虑按照 Lee 和 Shin (2023) 最近的工作，在两阶段过程中训练 DeepONet，其中第一阶段从主干网络中提取基础，然后对其进行正交归一化，然后将其用于训练分支网络的第二阶段。 DeepONet 的这种简单修改对其准确性、效率和鲁棒性产生了深远的影响，并且与普通版本相比，可以非常准确地解决黎曼问题。它还使我们能够以物理方式解释结果，因为分层数据驱动生成的基础反映了所有流特征，否则将使用临时特征扩展层引入这些特征。我们还将结果与另一个基于 U-Net 的神经算子进行了比较，对于低、中和非常高的压力比，这些算子对于黎曼问题非常准确，特别是对于大压力比，因为它们的多尺度性质，但计算成本更高。总的来说，我们的研究表明，简单的神经网络架构如果经过适当的预训练，可以实现黎曼问题的非常准确的解决方案以进行实时预测。源代码及其相应的数据可以在以下 URL 中找到： 。