Using weak constrained neural networks to improve simulations in the gray zone

<p>Machine learning represents a potential method to cope with the gray zone problem of representing motions in dynamical systems on scales comparable to the model resolution. Here we explore the possibility of using a neural network to directly learn the error caused by unresolved scales. We use a modified shallow water model which includes highly nonlinear processes mimicking atmospheric convection. To create the training dataset, we run the model in a high- and a low-resolution setup and compare the difference after one low-resolution time step, starting from the same initial conditions, thereby obtaining an exact target. The neural network is able to learn a large portion of the difference when evaluated on single time step predictions on a validation dataset. When coupled to the low-resolution model, we find large forecast improvements up to 1&#8201;d on average. After this, the accumulated error due to the mass conservation violation of the neural network starts to dominate and deteriorates the forecast. This deterioration can effectively be delayed by adding a penalty term to the loss function used to train the ANN to conserve mass in a weak sense. This study reinforces the need to include physical constraints in neural network parameterizations.</p>