RiemannONets: Interpretable Neural Operators for Riemann Problems

TL;DR

This work develops RiemannONets, a pair of neural-operator models for solving 1D Riemann problems in compressible Euler flows with extreme pressure jumps. It combines a DeepONet with a two-step training and a parameter-conditioned U-Net to map initial left-side pressures to final density, velocity, and pressure fields, achieving high accuracy across low to very high pressure ratios while enabling real-time inference. A key contribution is the interpretability of learned representations, using QR/SVD-based basis functions to reveal hierarchical, physically meaningful modes and to mitigate Gibbs oscillations near discontinuities. The study also assesses adaptive Rowdy activations and enforces positivity of density and pressure, providing practical design principles for robust, fast simulations of shock-dominated flows. Overall, RiemannONets demonstrates that simple neural architectures, properly pre-trained and analyzed, can provide accurate, real-time solvers for challenging discontinuous PDE problems with potential real-world impact in aerospace and fluid dynamics.

Abstract

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

Figures (15)

• Figure 1: Schematic representation of RiemannONet for the Sod's problem. The input to the trunk net is the sensor location in spatial dimensions, while the input to the branch net is different realizations of left side pressure (keeping right side pressure fixed). The output of RiemannONet is the primitive variables consisting of velocity, density, and pressure at the final time.
• Figure 2: U-Net conditioned on pressure initialization ($p_l$). $U^{ref} \subseteq U^{\text{train}}$ is provided as input to a U-Net which behaves like a multi-scale neural operator. The output of each encoder block, $\vec{z}^{\mathcal{L}_p}$, is conditioned on the parameter $p_l$ through an element-wise product operation. The corresponding representation is concatenated with the previous decoder block's output ($\vec{d}^{\mathcal{L}_{p-1}}_{p_l}$), and subsequently projected back as the output of the model.
• Figure 3: Low pressure ratio test case: Comparison of $\tanh$, ReLu, and Rowdy $\tanh$ adaptive activation functions. The density of four samples is inferred from the testing data set. The predictive accuracy of the Rowdy $\tanh$ adaptive activation functions is better than that of its fixed counterparts.
• Figure 4: Low Pressure Ratio Sod Problem: Comparison of DeepONet and U-Net results. The first row shows the DeepONet results for density, velocity, and pressure, whereas the second row shows the corresponding results for U-Net. The density of four samples is inferred from the testing data set.
• Figure 5: Intermediate Pressure Ratio Sod Problem: Comparison of DeepONet and U-Net results. The first row shows the DeepONet results for density, velocity, and pressure, whereas the second row shows the corresponding results for U-Net. The density of four samples is inferred from the testing data set.