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Seminar zu Machine Learning and Data Analytics

 

Üblicherweise erster Donnerstag im Monat, 16:00 Uhr, FZ.02.06, Gebäude FZ.

 

Bergische Universität Wuppertal, Campus Freudenberg, 42119 Wuppertal

 

Im Seminar werden aktuelle Forschungsthemen aus Wissenschaft- und Technikforschung in den Themengebieten maschinelles Lernen und Datenanalyse diskutiert.

ACHTUNG: Der aktuelle Vortrag am 05.05.2022 beginnt bereits um 12:00 und findet nur online via Zoom statt. Falls Sie noch keine Zugangsdaten haben, wenden Sie sich bitte an die Geschäftsstelle.

 

 

Datum Referent Vortrag
07.04.2022 Thomas Thiele, Data Intelligence Center - House of AI, Deutsche Bahn AG Heavy Metal meets AI – Künstliche Intelligenz in Mobilität und Logistik auf der Schiene
05.05.2022 Arnulf Jentzen, Chinese University of Hong Kong, Shenzhen & Universität Münster Overcoming the Curse of Dimensionality: from nonlinear Monte Carlo to Deep Learning
02.06.2022 Siniša Šegvić, University of Zagreb Elements of Learning Algorithms for Natural Scene Understanding
30.06.2022 Tim Fingscheidt, Technische Universität Braunschweig wird noch bekannt gegeben
07.07.2022 Sebastian Stober, Otto-von-Guericke-Universität Magdeburg Cognitive neuroscience inspired techniques for eXplainable AI (CogXAI)
Stand: 03.05.2022

Abstract for the current seminar:


"Overcoming the Curse of Dimensionality: from nonlinear Monte Carlo to Deep Learning"

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems.
For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schrödinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companies aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives.

The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model.
For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio.
Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs.

Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE.
With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used.
In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula.

In this talk we prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE can be solved approximatively without the curse of dimensionality.