Augmenting a Physics-Informed Neural Network for the 2D Burgers Equation by Addition of Solution Data Points

We implement a Physics-Informed Neural Network (PINN) for solving the two-dimensional Burgers equations. This type of model can be trained with no previous knowledge of the solution; instead, it relies on evaluating the governing equations of the system in points of the physical domain. It is also possible to use points with a known solution during training. In this paper, we compare PINNs trained with different amounts of governing equation evaluation points and known solution points. Comparing models that were trained purely with known solution points to those that have also used the governing equations, we observe an improvement in the overall observance of the underlying physics in the latter. We also investigate how changing the number of each type of point affects the resulting models differently. Finally, we argue that the addition of the governing equations during training may provide a way to improve the overall performance of the model without relying on additional data, which is especially important for situations where the number of known solution points is limited.

A Physics-Informed Neural Network to Model Port Channels

We describe a Physics-Informed Neural Network (PINN) that simulates the flow induced by the astronomical tide in a synthetic port channel, with dimensions based on the Santos - S\~ao Vicente - Bertioga Estuarine System. PINN models aim to combine the knowledge of physical systems and data-driven machine learning models. This is done by training a neural network to minimize the residuals of the governing equations in sample points. In this work, our flow is governed by the Navier-Stokes equations with some approximations. There are two main novelties in this paper. First, we design our model to assume that the flow is periodic in time, which is not feasible in conventional simulation methods. Second, we evaluate the benefit of resampling the function evaluation points during training, which has a near zero computational cost and has been verified to improve the final model, especially for small batch sizes. Finally, we discuss some limitations of the approximations used in the Navier-Stokes equations regarding the modeling of turbulence and how it interacts with PINNs.

Modeling Oceanic Variables with Dynamic Graph Neural Networks

Researchers typically resort to numerical methods to understand and predict ocean dynamics, a key task in mastering environmental phenomena. Such methods may not be suitable in scenarios where the topographic map is complex, knowledge about the underlying processes is incomplete, or the application is time critical. On the other hand, if ocean dynamics are observed, they can be exploited by recent machine learning methods. In this paper we describe a data-driven method to predict environmental variables such as current velocity and sea surface height in the region of Santos-Sao Vicente-Bertioga Estuarine System in the southeastern coast of Brazil. Our model exploits both temporal and spatial inductive biases by joining state-of-the-art sequence models (LSTM and Transformers) and relational models (Graph Neural Networks) in an end-to-end framework that learns both the temporal features and the spatial relationship shared among observation sites. We compare our results with the Santos Operational Forecasting System (SOFS). Experiments show that better results are attained by our model, while maintaining flexibility and little domain knowledge dependency.