Using Neural Networks to solve the Burgers Equation

Partial Differential Equations (PDEs) are differential equations that have multiple variables and one or more of their partial derivatives. However, this property makes writing explicit solutions for PDEs often impossible, and solutions such as numerical solvers can be expensive. One such example is the Burgers Equation, which models the behavior of viscous fluids, and has real world applications such as modeling turbulence. Neural Networks (NNs) are powerful machine learning models that simulate biological neural networks in animals with artificial neurons. NNs are capable of learning relationships from existing data, potentially providing fast and inexpensive solutions to PDEs. This project seeks to create a NN to solve the Burgers Equation for any spatial location and time given an initial condition. Using a PDEBench dataset with solution fields to the Burgers Equation, we created tensors for the training and evaluation of our model. Using the PyTorch library, a neural network with linear transformations and ReLU activation layers was created. After training the model, the true values were plotted against the predicted values, where a perfect linear fit meant the model was 100% accurate in finding solutions. Overall, the model was effective, showing a relatively linear relationship between the true and predicted values.