RiemannONets: Interpretable Neural Operators for Riemann Problems Ahmad Peyvan, Vivek Oommen, Ameya D. Jagtap, George Em Karniadakis This paper presents RiemannONets, a neural operator framework for solving Riemann problems in compressible flows with extreme pressure ratios. The method employs a modified DeepONet architecture, trained in a two-stage process, which enhances accuracy, efficiency, and robustness. The first stage extracts an orthonormal basis from the trunk net, which is then used in the second stage to train the branch net. This approach enables the interpretation of results as the hierarchical basis reflects all flow features. The method is compared with a U-Net-based neural operator, which is effective for multiscale problems but computationally more expensive. The study demonstrates that simple neural networks, when properly pre-trained, can achieve accurate solutions for Riemann problems in real-time forecasting. The source code and data are available at the provided URL. Keywords: Neural operator networks, Riemann problems, Compressible flows, DeepONet, U-Net, Data-driven basis The paper introduces two neural operators: DeepONet and U-Net. DeepONet is trained in a two-stage process, with the first stage extracting an orthonormal basis from the trunk net. The second stage trains the branch net using this basis. The U-Net is conditioned on initial pressure and temperature states. The study evaluates the performance of these operators for low, intermediate, and high-pressure ratios. The results show that the modified DeepONet provides accurate solutions, especially for high-pressure ratios, while the U-Net is effective for multiscale problems. The study also explores the use of adaptive activation functions and positivity preservation constraints to improve accuracy and stability. The results demonstrate that the modified DeepONet outperforms the U-Net in density prediction but is comparable in velocity and pressure prediction. The study also analyzes the hierarchical basis functions of the operators, showing that the SVD decomposition provides a more robust and interpretable basis than QR decomposition. The results highlight the effectiveness of the proposed method in solving Riemann problems with discontinuous solutions.RiemannONets: Interpretable Neural Operators for Riemann Problems Ahmad Peyvan, Vivek Oommen, Ameya D. Jagtap, George Em Karniadakis This paper presents RiemannONets, a neural operator framework for solving Riemann problems in compressible flows with extreme pressure ratios. The method employs a modified DeepONet architecture, trained in a two-stage process, which enhances accuracy, efficiency, and robustness. The first stage extracts an orthonormal basis from the trunk net, which is then used in the second stage to train the branch net. This approach enables the interpretation of results as the hierarchical basis reflects all flow features. The method is compared with a U-Net-based neural operator, which is effective for multiscale problems but computationally more expensive. The study demonstrates that simple neural networks, when properly pre-trained, can achieve accurate solutions for Riemann problems in real-time forecasting. The source code and data are available at the provided URL. Keywords: Neural operator networks, Riemann problems, Compressible flows, DeepONet, U-Net, Data-driven basis The paper introduces two neural operators: DeepONet and U-Net. DeepONet is trained in a two-stage process, with the first stage extracting an orthonormal basis from the trunk net. The second stage trains the branch net using this basis. The U-Net is conditioned on initial pressure and temperature states. The study evaluates the performance of these operators for low, intermediate, and high-pressure ratios. The results show that the modified DeepONet provides accurate solutions, especially for high-pressure ratios, while the U-Net is effective for multiscale problems. The study also explores the use of adaptive activation functions and positivity preservation constraints to improve accuracy and stability. The results demonstrate that the modified DeepONet outperforms the U-Net in density prediction but is comparable in velocity and pressure prediction. The study also analyzes the hierarchical basis functions of the operators, showing that the SVD decomposition provides a more robust and interpretable basis than QR decomposition. The results highlight the effectiveness of the proposed method in solving Riemann problems with discontinuous solutions.