Optimizing finite-element mesh construction for singularly perturbed differential equations using neural networks

We use differential equations as a mathematical tool to enable us to model real world systems. This thesis focuses on developing new tools to help us understand models based on differential equations using numerical approximation. Instead of developing new approximation techniques, we focus on refining the inputs to these approximation tools by optimizing the mesh points given to them. To do this, we develop neural network algorithms to optimize mesh points for a given differential equation, utilizing both the exact solution to a problem and a higher order approximation as comparison tools for our loss function. This work was done by combining the Python packages Firedrake and Pytorch. Firedrake was used to generate the numerical approximations for us, and allowed us to easily try different permutations of problems and use higher order approximations when needed. Firedrake also allowed us to use finite-element discretizations without having to manually rediscretize our problem if we wanted to change something. Pytorch was the software used to create our neural networks that allowed us to generate meshes that over time would adapt to better approximate the solution of the desired differential equation. With these tools, we show that neural networks are a good resource for optimizing mesh points for a given differential equation without knowing the solution to the problem already.