Michael Hintermüller

Martin Burger, Yiqiu Dong, Michael Hintermüller

Relying on the co-area formula, an exact relaxation framework for minimizing objectives involving the total variation of a binary valued function (of bounded variation) is presented. The underlying problem class covers many important applications ranging from binary image restoration, segmentation, minimal compliance topology optimization to the optimal design of composite membranes and many more. The relaxation approach turns the binary constraint into a box constraint. It is shown that thresholding a solution of the relaxed problem almost surely yields a solution of the original binary-valued problem. Furthermore, stability of solutions under data perturbations is studied, and, for applications such as structure optimization, the inclusion of volume constraints is considered. For the efficient numerical solution of the relaxed problem, a locally superlinearly convergent algorithm is proposed which is based on an exact penalization technique, Fenchel duality, and a semismooth Newton approach. The paper ends by a report on numerical results for several applications in particular in mathematical image processing.

Michael Hintermüller, Michael Hinze, Denis Korolev

In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls acting as coefficients and a specific loss function. We implement our approach on the basis of a DG(0) Galerkin discretization of the neural ODE, leading to an explicit Euler time marching scheme. For the optimization we use steepest descent. Finally, we apply our method to the construction of neural networks for the classification of data sets, where we present results for a selection of well known examples from the literature.

Denis Korolev, Tim Schmidt, Dinesh K. Natarajan et al.

This study introduces a hybrid machine learning-based scale-bridging framework for predicting the permeability of fibrous textile structures. By addressing the computational challenges inherent to multiscale modeling, the proposed approach evaluates the efficiency and accuracy of different scale-bridging methodologies combining traditional surrogate models and even integrating physics-informed neural networks (PINNs) with numerical solvers, enabling accurate permeability predictions across micro- and mesoscales. Four methodologies were evaluated: Single Scale Method (SSM), Simple Upscaling Method (SUM), Scale-Bridging Method (SBM), and Fully Resolved Model (FRM). SSM, the simplest method, neglects microscale permeability and exhibited permeability values deviating by up to 150\% of the FRM model, which was taken as ground truth at an equivalent lower fiber volume content. SUM improved predictions by considering uniform microscale permeability, yielding closer values under similar conditions, but still lacked structural variability. The SBM method, incorporating segment-based microscale permeability assignments, showed significant enhancements, achieving almost equivalent values while maintaining computational efficiency and modeling runtimes of ~45 minutes per simulation. In contrast, FRM, which provides the highest fidelity by fully resolving microscale and mesoscale geometries, required up to 270 times more computational time than SSM, with model files exceeding 300 GB. Additionally, a hybrid dual-scale solver incorporating PINNs has been developed and shows the potential to overcome generalization errors and the problem of data scarcity of the data-driven surrogate approaches. The hybrid framework advances permeability modelling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing.