RICAM Colloquium - Philipp Petersen, Numerical Analysis for Deep Learning and Deep Learning for Numerical Analysis
Thursday, May 23, 2024, 15:30, SP 416-2
Deep learning has been a very successful tool in the last decade. This success has inspired applied mathematicians to attempt to use deep learning tools in applications that were traditionally served by numerical analysis. While this has been successful at times, deep learning tools often fall short in comparison to classical approaches, such as finite element-based solutions of PDEs, because they lack formal guarantees and rarely achieve high accuracies. In this talk, we study two examples that underscore this general trend. First, we find a class of problems---learning of high-dimensional functions with structured
singularities--- where deep learning clearly outperforms classical methods. Second, we observe that training of deep neural networks is numerically so unstable that high accuracies cannot be achieved. We conclude that merging ideas from numerical analysis and deep learning is necessary to advance both fields.