Termine Angewandte Mathematik Münster

The application of neural networks to solve nonlinear partial differential equations (PDEs) has gained significant attention, with popular methods including Physics-Informed Neural Networks (PINNs) and the deep-Ritz framework. However, these methods face challenges when applied to high-dimensional or instationary problems. A promising alternative is the use of evolutionary deep neural networks (EDNNs), which allow the network parameters to evolve over time. After the network is fitted to the initial conditions, the parameter evolution is then governed by an ordinary differential equation (ODE), solvable with established numerical integration schemes. Building upon this concept, the Neural Galerkin framework and subsequent publications provide further advances. The talk will include the derivation and mathematical interpretation of EDNNs, as well as extensions to parameter-dependent problems. Additionally, we will discuss the JAX-library used for efficient implementation of the proposed methods and present recent progress in goal-oriented error estimation for nonlinear discretizations.

Multitytype Galton-Watson processes in random environment (for short, MGWREs) are variants on the well known Galton-Watson process, where the offpsring distribution of an individual of the n-th generation depends both on a notion of type assigned to each individual, and on the value at time n of a stochastic process which represents the environment in which the population evolves. The study of MGWRE is closely related with some products of random matrices or linear operators , depending on whether the set of the possible types of the individuals is finite or not. In this talk, I will present some historical ergodicity results for products of random matrices, as well as an extension of these results to products of infinite dimensional operators. Moreover, I will explain how these ergodicity results allow to characterize the extinction of the associated Galton-Watson process and obtain Kesten-Stigum-type theorems, which yields an asymptotic description of the population when it survives.

Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided non-intrusive surrogate modeling is being actively researched. In this talk, we present two methods in the scope of non-intrusive surrogate modeling. The first method regards extrapolation capability of surrogate models in the time domain. We propose a deep learning framework which generalizes well in thewhole time interval [0, T], when the high-fidelity training data is available only in a training time interval [0, T_0], with T_0 < T. The framework is composed of a convolutional autoencoder (CAE), the kernel dynamic mode decomposition (KDMD) method and a feed-forward neural network (FFNN). The KDMD is employed to evolve the dynamics of the latent space generated by the encoder part of the CAE. The original high-fidelity data set is then augmented with the KDMD-decoder-extrapolated data. We train the CAE along with the FFNN using the augmented data. The trained network yields all-at-once parameter-time sequence prediction at any unseen testing parameters and at any future times t ? [T_0, T], T > T_0. This approach differs from the auto-regressive prediction methods used in existing works. The proposed method is tested on two numerical examples: a FitzHugh-Nagumo model and a model of flow past a cylinder. Numerical results show accurate and fast prediction performance in both the time and the parameter domain. The second method considers active learning for problems with high-dimensional parameter spaces. The training data generation for machine learning (ML) often takes considerable computational or experimental time, and is considered as the most expensive part of ML. This becomes especially time-consuming for problems with high-dimensional parameter spaces, where the training data increases exponentially with the number of samples in each parameter dimension. We propose an active learning technique for generating the training data only on demand, so that the finally trained NN requires much less training data and still achieves good accuracy, as compared to NN training without active learning. This technique is combined with CAE and FFNN to construct a surrogate model in MEMS design with many design parameters. A MEMS actuator model with 25 design parameters is tested to show the efficiency of the proposed technique.

We study the continuous limit of minimum-cost flow problems on large random graphs. Our main result asserts the convergence of minimum-cost flows to currents minimising a continuous linear-growth functional. Due to the particular growth regime, one of the main features and challenges is the presence of finite-energy flows with singularities. In this talk, we discuss the main ideas and tools to handle this framework, in particular the subadditive ergodic theorem and the blow-up method.

Dieses Praktikum richtet sich vor allem an Studierende der Mathematik, Physik, Chemie und Biologie. In interdisziplinären Kleingruppen werden anwendungsorientierte Fragestellungen bearbeitet.

Viele physikalische, chemische oder auch biologische Prozesse beinhalten mehrere zeitliche und räumliche Skalen. Die zugrunde liegenden Prozesse weisen zudem häufig eine nichtlineare Dynamik auf, die das makroskopische Verhalten maßgeblich bestimmt. Die Modellierung solcher Prozesse stellt aufgrund der komplexen Dynamik eine große Herausforderung dar, sowohl bei der physikalischen, chemischen oder biologischen Formulierung als auch der mathema- tischen/numerischen Behandlung.

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