Error Analysis and Optimization Methods for Neural Network Based PDE solvers
- Speaker: Dr. Marius Zeinhofer
- Place: 1.067
- Time: 14.5.2024, 13:30
- Invited by: Prof. Dr. Dorothee Frey
Abstract
In the first part of the talk, we discuss error estimates for neural network based PDE solution methods, such as physics-informed neural networks (PINNs) and the deep Ritz method. For the analysis, we propose an abstract framework in the language of bilinear forms, and we show the required continuity and coercivity estimates for the mentioned equations. For PINNs, our results reveal that the L2 penalty approach that is commonly employed for boundary and initial conditions leads to a pronounced deterioration in convergence mode. In the second part, we focus on optimization methods that can be derived from an infinite-dimensional viewpoint. More precisely, we will discretize well-known function space algorithms (such as Newton's method) in the tangent space of a neural network ansatz class and show that they lead to highly effective methods in practice.