On the Stable Integration of Neural Network Parameterization in Numerical Models

Deep learning-based parameterizations of subgrid-scale processes have become a major research focus in recent years, offering the potential to remedy inaccuracies inherent in traditional physics-based schemes. However, their integral stability within numerical models remains insufficiently explored. In this study, we develop deep learning parameterizations for the tropical cyclone boundary layer and implement them in the WRF model. We find that one-dimensional convolutional neural network fails to integrate stably, whereas a fully connected network succeeds. Further analysis shows that the limited receptive field of the convolutional network makes its outputs overly sensitive to certain input perturbations, ultimately causing integral instability. We examine three stabilization strategies—training data augmentation with Gaussian noise, spectral norm regularization, and L2 regularization—and find that all three methods effectively mitigate the network’s output sensitivity to input perturbations, enabling stable integration in WRF and yielding physically reasonable tropical cyclone simulations.