Data-Driven Stabilization Schemes for Singularly Perturbed Partial Differential Equations

Sangeeta Yadav
Indian Institute of Science
Tuesday 18th April 2023
In this talk, the speaker will introduce innovative ways to use Artificial Neural Networks (ANN) in conjunction with conventional numerical techniques to solve challenging Singularly Perturbed Differential Equations (SPDEs). SPDEs often exhibit spurious oscillations in the standard numerical solution due to the presence of boundary and interior layers. Stabilization techniques are employed to eliminate these oscillations, but the accuracy of these techniques depends on a user-selected stabilization parameter that is difficult to determine optimally. To address this challenge, the authors have developed several ANN-based techniques to predict the optimal stabilization parameter. The first method, called SPDE-Net, uses ANN to predict the stabilization parameter for the streamline upwind/Petrov-Galerkin (SUPG) stabilization technique for solving one-dimensional SPDEs. The second method, AI-stab FEM, is an optimization scheme that employs an unsupervised ANN to minimize the residual along with the crosswind derivative term to solve two-dimensional SPDEs. The third method is called the SPDE-ConvNet, a convolutional neural network that predicts the local (cell-wise) stabilization parameter. The proposed techniques have outperformed other state-of-the-art ANNbased partial differential equation (PDE) solvers such as Physics Informed Neural Network (PINN) and VarNet for several benchmark problems. In conclusion, the speaker will compare the performance of these data-driven stabilization schemes and discuss potential future extensions. References: [1] S. Yadav, S. Ganesan, SPDE-Net: Neural Network based prediction of stabilization parameter for SUPG technique Proceedings of the 13th Asian Conference on Machine Learning, PMLR 157:268-283, (2021) [2] S. Yadav, S. Ganesan, AI-augmented stabilized finite element method. arXiv:2211.13418 (2022). [3] S. Yadav, S. Ganesan, SPDE-ConvNet: Solve Singularly Perturbed Partial Differential Equation with deep learning. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS CONGRESS (2022).