  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
