Aktuelles & Events

New deep neural network architectures based on high-order weak approximation algorithms for stochastic differential equations (SDEs) are proposed. The core of these architectures is formed by high-order weak approximation algorithms of the explicit Runge--Kutta type, in which the approximation is realised solely through iterative compositions and linear combinations of the vector fields of the target SDEs.

In my talk I will consider the following question: how many random hyperplanes are needed to uniformly tessellate a given subset of Rn with high probability? In my talk I will present an optimal answer to this question for selected distributions for the random hyperplanes and sketch three applications of these results in the mathematical foundations of data science. First, I will show how to create a fast encoding of any given dataset into a minimal number of bits. Second, I will consider performance guarantees for one-bit compressed sensing methods, which aim to reconstruct a signal from a small number of measurements that are each quantized to a single bit using an efficient analog-to-digital converter. Third, I will discuss implications for the robustness of ReLU neural networks. The talk will be a survey-style presentation for a general mathematical audience. Based on joint works with Shahar Mendelson (ANU Canberra), Alexander Stollenwerk (Louvain), Patrick Finke, Nigel Strachan (Utrecht), Paul Geuchen, Dominik Stöger, Felix Voigtlaender (Eichstätt-Ingolstadt) ___________________________ Invited by Prof. Johannes Maly

We discuss recent developments around Hilbert's Sixth Problem about the passage from kinetic models to macroscopic fluid equations. We employ the technique of slow spectral closure to rigorously establish the existence of hydrodynamic manifolds in the linear regime and derive new non-local fluid equations for rarefied flows independent of Knudsen number. We show the divergence of the Chapman--Enskog series for an explicit example and apply machine learning to learn the optimal hydrodynamic closure from DSMC and SBGK data. The new dynamically optimal constitutive laws are applied to classical rarefied flow problems and the light scattering experiment.

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