Marius Zeinhofer - Error Analysis and Optimization Methods for Scientific Machine Learning 

Abstract: In the first part of the talk, we discuss error estimates for physics-informed neural networks (PINNs) for a wide range of linear PDEs, including elliptic, parabolic and hyperbolic equations. 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. Our results illustrate that the L2 penalty approach that is commonly employed for boundary and initial conditions provably leads to a pronounced deterioration in convergence mode. In the second part, we focus on optimization methods for scientific machine learning 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.