Główne pojęcia
Neural operators, including DeepONet and U-Net, can accurately solve Riemann problems with extreme pressure jumps up to 10^10 pressure ratio.
Streszczenie
The authors investigate the use of neural operators, specifically DeepONet and U-Net, to solve Riemann problems encountered in compressible flows. They consider three test cases with low, intermediate, and high-pressure ratios, including the challenging LeBlanc problem with pressure ratios up to 10^10.
Key highlights:
The DeepONet architecture is modified to include a two-stage training process, where the first stage extracts a basis from the trunk net, which is then used in the second stage to train the branch net. This leads to improved accuracy, efficiency, and robustness compared to the vanilla DeepONet.
The U-Net architecture is conditioned on the initial pressure and temperature states, enabling it to capture the multiscale nature of the solutions, particularly for large pressure ratios.
The authors analyze the hierarchical and interpretable basis functions produced by the neural operators, providing insights into the representation of the discontinuous solutions.
The results demonstrate that the simple neural network architectures, if properly pre-trained, can achieve very accurate solutions of Riemann problems for real-time forecasting.
Statystyki
The pressure ratio ranges from 1.008 to 3.96 for the low-pressure ratio case, 51.9 to 88.7 for the intermediate-pressure ratio case, and 1.342 × 10^9 to 7.966 × 10^9 for the high-pressure ratio (LeBlanc) case.
Cytaty
"Our study leverages the capabilities of deep neural operators to investigate its efficacy in mapping input pressure ratios to the final solution at a specified time."
"We obtain interpretable basis functions for such discontinuous solutions. To this end, we employ QR and SVD methods to investigate the solution spectrum and diverse bases."
"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."