Georgia Institute of Technology College of Sciences

The use of neural networks for solving partial differential equations (PDEs) has attracted considerable attention in recent years. In this talk, I will first highlight their advantages over traditional numerical methods, including improved approximation rates and the potential to overcome the curse of dimensionality. I will then discuss the challenges that arise when applying neural networks to PDEs, particularly in training. Because training is inherently a highly nonconvex optimization problem, it can lead to poor local minima with large training errors, especially in complex PDE settings. To address these issues, I will demonstrate how incorporating mathematical insight into the design of training algorithms and network architectures can lead to significant improvements in both accuracy and robustness.