## Christoph Schwab: Catalogue data in Autumn Semester 2018 |

Name | Prof. Dr. Christoph Schwab |

Field | Mathematik |

Address | Seminar für Angewandte Mathematik ETH Zürich, HG G 57.1 Rämistrasse 101 8092 Zürich SWITZERLAND |

Telephone | +41 44 632 35 95 |

Fax | +41 44 632 10 85 |

christoph.schwab@sam.math.ethz.ch | |

URL | http://www.sam.math.ethz.ch/~schwab |

Department | Mathematics |

Relationship | Full Professor |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-3650-68L | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Number of participants limited to 6. | 4 credits | 2S | C. Schwab | |

Abstract | This seminar will review recent (2016-) _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. | ||||

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 application areas (Chess, Go, autonomous driving, language translation, image classification, etc.). In many cases, these successes have been achieved by implementations, based on heuristics, with massive compute power and training data. This seminar will review recent (2016-) _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. Also here, there is mounting mathematical evidence that DNNs equalize or outperform the best known mathematical results. 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. The format will be oral student presentations in December 2018 based on a recent research paper selected in two meetings at the start of the semester. | ||||

Literature | Partial reading list: arXiv:1809.07669 DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing Authors: Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen, Christoph Schwab arXiv:1806.08459 Topological properties of the set of functions generated by neural networks of fixed size Authors: Philipp Petersen, Mones Raslan, Felix Voigtlaender arXiv:1804.10306 Universal approximations of invariant maps by neural networks Author: Dmitry Yarotsky arXiv:1802.03620 Optimal approximation of continuous functions by very deep ReLU networks Author: Dmitry Yarotsky arXiv:1709.05289 Optimal approximation of piecewise smooth functions using deep ReLU neural networks Authors: Philipp Petersen, Felix Voigtlaender arXiv:1706.03301 Neural networks and rational functions Author: Matus Telgarsky arXiv:1705.05502 The power of deeper networks for expressing natural functions Authors: David Rolnick, Max Tegmark arXiv:1705.01365 Quantified advantage of discontinuous weight selection in approximations with deep neural networks Author: Dmitry Yarotsky arXiv:1610.01145 Error bounds for approximations with deep ReLU networks Author: Dmitry Yarotsky arXiv:1608.03287 Deep vs. shallow networks : An approximation theory perspective Authors: Hrushikesh Mhaskar, Tomaso Poggio arXiv:1602.04485 Benefits of depth in neural networks Author: Matus Telgarsky | ||||

Prerequisites / Notice | 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, such as training heuristics, or programming techniques for various specific applications. | ||||

401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics | 0 credits | 2K | R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, A. Jentzen, C. Jerez Hanckes, S. Mishra, S. Sauter, C. Schwab | |

Abstract | Research colloquium | ||||

Objective |