Tom Freudenberg

Derick Nganyu Tanyu, Jianfeng Ning, Tom Freudenberg et al.

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

Tom Freudenberg, Michael Eden

We investigate the effective coupling between heat and fluid dynamics within a thin fluid layer in contact with a solid structure via a rough surface. Moreover, the opposing vertical surfaces of the thin layer are in relative motion. This setup is particularly relevant to grinding processes, where cooling lubricants interact with the rough surface of a rotating grinding wheel. The resulting model is non-linearly coupled through(i) temperature-dependent viscosity and (ii) convective heat transport. The underlying geometry is highly heterogeneous due to the thin, rough surface characterized by a small parameter representing both the height of the layer and the periodicity of the roughness. We analyze this non-linear system for existence, uniqueness, and energy estimates and study the limit behavior within the framework of two-scale convergence in thin domains. In this limit, we derive an effective interface model in 3D (a line in 2D) for the heat and fluid interactions inside the fluid. We implement the system numerically and validate the limit problem through direct comparison with the micromodel. Additionally, we investigate the influence of the temperature-dependent viscosity and various geometrical configurations via simulation experiments. The corresponding numerical code is freely available on GitHub.

Alexander Denker, Zeljko Kereta, Imraj Singh et al.

Electrical impedance tomography (EIT) plays a crucial role in non-invasive imaging, with both medical and industrial applications. In this paper, we present three data-driven reconstruction methods for EIT imaging. These three approaches were originally submitted to the Kuopio tomography challenge 2023 (KTC2023). First, we introduce a post-processing approach, which achieved first place at KTC2023. Further, we present a fully learned and a conditional diffusion approach. All three methods are based on a similar neural network as a backbone and were trained using a synthetically generated data set, providing with an opportunity for a fair comparison of these different data-driven reconstruction methods.