# 401-3650-68L Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation

Semester | Autumn Semester 2019 |

Lecturers | C. Schwab |

Periodicity | yearly recurring course |

Language of instruction | English |

Comment | Number of participants limited to 6. Consent of Instructor needed. |

### Courses

Number | Title | Hours | Lecturers | |
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401-3650-00 S | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Permission from lecturers required for all students.
Preliminary discussions and assignment of seminar topic to participants: Monday, 16 September 2019 at 13:15 in HG E 33.1. Student talks are planned to take place on .... The room reservations will be announced in due course. | 2 hrs | C. Schwab |

### Catalogue data

Abstract | The 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. |

Objective | |

Content | 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 big data analysis, DNNs achieved remarkable performance in computer vision, speech recognition and natural language processing. In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. For a (bird's eye) view, see https://arxiv.org/abs/1901.05639 and, more mathematical and closer to the seminar theme, https://arxiv.org/abs/1901.02220 The 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. Format of the Seminar: The seminar format will be oral student presentations, combined with written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. Grading of the Seminar: Passing grade will require a) 1hr oral presentation with Q/A from the seminar group and b) typed seminar report (``Ausarbeitung'') of several key aspects of the paper under review. Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. Disclaimer: The seminar will _not_ address recent developments in DNN software, eg. TENSORFLOW, and algorithmic training heuristics, or programming techniques for DNN training in various specific applications. |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 4 credits |

Examiners | C. Schwab |

Type | ungraded semester performance |

Language of examination | English |

Repetition | Repetition only possible after re-enrolling for the course unit. |

Admission requirement | Completed BSc MATH ETH and Consent of Instructor |

Additional information on mode of examination | Passing grade will require a) 1hr oral presentation with Q/A from the seminar group and b) typed seminar report (``Ausarbeitung'') of several key aspects of the paper under review. |

### Learning materials

No public learning materials available. | |

Only public learning materials are listed. |

### Groups

No information on groups available. |

### Restrictions

General | Permission from lecturers required for all students |

Places | 6 at the most |

Priority | Registration for the course unit is only possible for the primary target group |

Primary target group | Mathematics MSc (437000)
Applied Mathematics MSc (437100) Computational Science and Engineering MSc (438000) |

Waiting list | until 19.09.2019 |

### Offered in

Programme | Section | Type | |
---|---|---|---|

Mathematics Master | Seminars | W |