He Jingjin, Lu Changna, Miao Yuxing
This paper proposes a Convolutional Neural Network (CNN)-based WENO scheme, named WENO-CNN, which aims to optimize the nonlinear weights in the WENO framework through machine learning techniques to enhance the accuracy and robustness of numerical simulations. Traditional WENO-JS schemes address discontinuities by weighting different reconstruction polynomials. To enable weight adaptation to diverse flow conditions, WENO-CNN employs a CNN to predict weights at identified troubled cells, allowing dynamic weight adjustments based on flow field characteristics. This approach improves numerical performance in scenarios involving strong discontinuities and complex flow structures. A specialized CNN architecture is developed for flow field problems, incorporating residual layers with skip connections. The network design includes an input layer that captures essential flow field features, ReLU activation functions, a customized loss function, and the Adam optimizer for training. Extensive validation through five classical test cases governed by the Euler equations demonstrates that the WENO-CNN scheme achieves higher shock wave resolution with refined detail capture while maintaining stability in smooth regions. The scheme effectively suppresses numerical oscillations, showing significant improvements over conventional methods in both shock and discontinuity flow simulations.
