The finite element method (FEM) is a numerical method that is used to approximate the solutions to partial differential equations when solutions in the classical sense do not exist or are very hard to find. The method is used to solve problems that are relevant for industries like the automotive industry, the petroleum industry, and the aviation industry. The finite element method uses a discretisation of the problem domain into sub domains and uses a variational form involving trial and test functions to arrive at a system of equations. For problems that are advection dominated (for problems with a large Péclet number), the FEM solutions start to oscillate and produce inaccurate results near boundary layers. In the past finite element schemes have been developed that use so called optimal test functions to reduce the oscillations of the solution and increase the accuracy of the method.
In this thesis an attempt was made to use Deep Operator Networks (DeepONets) to generate optimal test functions for the steady state advection-diffusion equation in 1D and 2D to improve the stability/accuracy of the finite element method. An advantage of using neural networks is that once trained they can take in problem specific parameters like the diffusion coefficient and produce optimal test functions for a wide range of problems almost instantaneously. It was found that the applicability of the DeepONets in this context varies and depends on the weak formulation for which the DeepONet generated optimal test functions are implemented, as the finite element method solution can be very sensitive to small perturbations in the optimal test functions. The approach does look promising as a DeepONet was able to improve the finite element method significantly in a 2D setting by generating optimal test functions, while using the problem specific diffusion coefficient as a variable.