# 401-3650-19L Numerical Analysis Seminar: Deep Neural Network Approximation

Semester | Frühjahrssemester 2021 |

Dozierende | C. Schwab |

Periodizität | jährlich wiederkehrende Veranstaltung |

Lehrveranstaltung | Findet dieses Semester nicht statt. |

Lehrsprache | Englisch |

Kurzbeschreibung | This seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. |

Lernziel | |

Inhalt | Presentation of the Seminar: Deep Neural Networks (DNNs) have recently attracted substantial interest and attention due to outperforming the best established techniques in a number of tasks (Chess, Go, Shogi, autonomous driving, language translation, image classification, etc.). In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. The seminar will address mathematical results on the approximation/ expressive power of DNNs. For a (bird's eye) overview, see Link and, more mathematical and closer to the seminar theme, Link Specifically, this seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. Mathematical results support that DNNs can equalize or outperform the best mathematical results known to date. Particular cases comprise: high-dimensional parametric maps, analytic and holomorphic maps, maps containing multi-scale features which arise as solution classes from PDEs, classes of maps which are invariant under group actions. |

Voraussetzungen / Besonderes | Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. The seminar format will be oral student presentations in the first half of May 2021, combined with a written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester (end of February). Disclaimer: The seminar will _not_ address recent developments in DNN software, such as training heuristics, or programming techniques for DNN training in various specific applications. |