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 $10^{10}$ pressure ratio). In particular, we first consider the DeepONet that we train in a two-stage process, following the recent work of \cite{lee2023training}, 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: https://github.com/apey236/RiemannONet/tree/main

翻译：如何构建合适的表示方法来模拟包含强激波、稀疏波和接触间断的高速流动，一直是数值分析领域长期悬而未决的问题。本文采用神经算子求解可压缩流中极端压力跃变（压力比高达$10^{10}$）下的黎曼问题。具体而言，我们首先采用DeepONet，并遵循\cite{lee2023training}的最新工作，通过两阶段流程进行训练：第一阶段从主干网络中提取基函数，对其进行正交归一化处理；第二阶段利用该基函数训练分支网络。这种对DeepONet的简单改进对其精度、效率和鲁棒性产生了深远影响，相较于原始版本，能够获得黎曼问题的高精度解。同时，该改进使我们能够从物理角度解释结果，因为层次化数据驱动生成的基函数反映了所有流动特征，而这些特征原本需通过特定特征扩展层引入。此外，我们将结果与基于U-Net的另一种神经算子进行对比，该算子在中、低及极高压力比下对黎曼问题均具有高精度——得益于其多尺度特性，尤其在大压力比下表现优异，但计算成本更高。总体而言，本研究表明，简单的神经网络架构若经过恰当预训练，即可实现黎曼问题的高精度实时预测。源代码及对应数据可从以下网址获取：https://github.com/apey236/RiemannONet/tree/main