Convolutional Neural network for local stabilization parameter prediction for Singularly Perturbed PDEs

Singularly Perturbed Partial Differential Equations are challenging to solve with conventional numerical techniques such as Finite Element Methods due to the presence of boundary and interior layers. Often the standard numerical solution has spurious oscillations in the vicinity of these layers. Stabilization techniques are employed to eliminate these spurious oscillations in the numerical solution. The accuracy of the stabilization technique depends on a user-chosen stabilization parameter, where an optimal value is challenging to find. In this work, we focus on predicting an optimal value of the stabilization parameter for a stabilization technique called the Streamline Upwind Petrov Galerkin technique for solving singularly perturbed partial differential equations. This paper proposes \textit{SPDE-ConvNet}, a convolutional neural network for predicting stabilization parameters by minimizing a loss based on the cross-wind derivative term. The proposed technique is compared with the state-of-the-art variational form-based neural network schemes.