## Further research projects of Research Area C members

$\bullet$ **Mario Ohlberger**, **Felix Schindler**: ML-MORE: Machine learning and model order reduction to predict the efficiency of catalytic filters. Subproject 1: Model Order Reduction (2020-2023)

Reaktiver Stofftransport in porösen Medien in Verbindung mit katalytischen Reaktionen ist die Grundlage für viele industrielle Prozesse und Anlagen, wie z.B. Brennstoffzellen, Photovoltaikzellen, katalytische Filter für Abgase, etc. Die Modellierung und Simulation der Prozesse auf der Porenskala kann bei der Optimierung des Designs von katalytischen Komponenten und der Prozessführung helfen, ist jedoch derzeit dadurch eingeschränkt, dass solche Simulationen zu grossen Datenmengen führen, zeitaufwändig sind und von einer grossen Anzahl von Parametern abhängen. Außerdem werden auf diese Weise die im Laufe der Jahre gesammelten Versuchsdaten nicht wiederverwendet. Die Entwicklung von Lösungsansätzen für die Vorhersage der chemischen Konversionsrate mittels moderner datenbasierter Methoden des Maschinellen Lernens (ML) ist essenziell, um zu schnellen, zuverlässigen prädiktiven Modellen zu gelangen. Hierzu sind verschiedene Methodenklassen erforderlich. Neben den experimentellen Daten sind voll aufgelöste Simulationen auf der Porenskala notwendig. Diese sind jedoch zu teuer, um einen umfangreichen Satz an Trainingsdaten zu generieren. Daher ist die Modellordnungsreduktion (MOR) zur Beschleunigung entscheidend. Es werden reduzierte Modelle fur den betrachteten instationären reaktiven Transport entwickelt, um große Mengen an Trainingsdaten zu simulieren. Als ML-Methodik werden mehrschichtige Kern-basierte Lernverfahren entwickelt, um die heterogenen Daten zu kalibrieren und nichtlineare prädiktive Modelle zur Effizienzvorhersage zu entwickeln.Hierbei werden große Daten (bzgl. Dimensionalität und Sample-Zahl) zu behandeln sein, was Datenkompression und Parallelisierung des Trainings erfordern wird. Das Hauptziel des Projekts ist es, alle oben genannten Entwicklungen in einem prädiktiven ML-Tool zu integrieren, das die Industrie bei der Entwicklung neuer katalytischer Filter unterstützt und auf viele andere vergleichbare Prozesse übertragbar ist.

$\bullet$ **Christian Seis**: Transport Equations, mixing and fluid dynamics (2020-2023)

Advection-diffusion equations are of fundamental importance in many areas of science. They describe systems, in which a quantity is simultaneously diffused and advected by a velocity field. In many applications these velocity fields are highly irregular. In this project, several quantitative aspects shall be investigated. One is related to mixing properties in fluids caused by shear flows. The interplay between the transport by the shear flow and the regularizing diffusion leads after a certain time, to the emergence of a dominant length scales which persist during the subsequent evolution and determine mixing rates. A rigorous understanding of these phenomena is desired. In addition, stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on coefficients and data. The new results shall subsequently be applied to estimate the error generated by numerical finite volume schemes approximating the model equations.

$\bullet$ **Raimar Wulkenhaar**: GRK 2149 - Starke und schwache Wechselwirkung - von Hadronen zu Dunkler Materie (2020-2024)

The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group.Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC.The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics.

$\bullet$ **Caterina Zeppieri**: Homogenisation and elliptic approximation of random free-discontinuity functionals (2020-2022)

Composite materials posses an incredibly complex microstructure. To reduce this complexity, in materials modelling reasonable idealizations have to be considered. Random composite materials represent a relevant class of such idealizations. Motivated by primary questions arising in the variational theory of fracture, the goal of this project is to study the large-scale behavior of random elastic composites which can undergo fracture. Mathematically this amounts to develop a qualitative theory of stochastic homogenization for free-discontinuity functionals. This will be done by combining two approaches: a "direct" approach and an "indirect" approximation-approach. The direct approach consists in extending the classical theory to the BV-setting. The approximation-approach, instead, consists in proposing suitable elliptic phase-field approximations of random free-discontinuity functionals which can provide regular-approximations of the homogenized coefficients.

$\bullet$ **Benedikt Wirth**: Mathematische Rekonstruktion und Modellierung der CAR T-Zell Verteilung in vivo in einem Tumormodell (2019-2023)

$\bullet$ **Arnulf Jentzen**: Mathematical Theory for Deep Learning (2019-2024)

It is the key goal of this project to provide a rigorous mathematical analysis for deep learning algorithms and thereby to establish mathematical theorems which explain the success and the limitations of deep learning algorithms. In particular, this projects aims (i) to provide a mathematical theory for high-dimensional approximation capacities for deep neural networks, (ii) to reveal suitable regular sequences of functions which can be approximated by deep neural networks but not by shallow neural networks without the curse of dimensionality,
and (iii) to establish dimension independent convergence rates for stochastic gradient descent optimization algorithms when employed to train deep neural networks with error constants which grow at most polynomially in the dimension.

$\bullet$ **Arnulf Jentzen**: Existence, uniqueness, and regularity properties of solutions of partial differential equations (2019-2024)

The goal of this project is to reveal existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs). In particular, we intend to study existence, uniqueness, and regularity properties of viscosity solutions of degenerate semilinear Kolmogorov PDEs of the parabolic type. We plan to investigate such PDEs by means of probabilistic representations of the Feynman-Kac type. We also intend to study the connections of such PDEs to optimal control problems.

$\bullet$ **Arnulf Jentzen**: Regularity properties and approximations for stochastic ordinary and
partial differential equations with non-globally Lipschitz continuous
nonlinearities (2019-2024)

A number of stochastic ordinary and partial differential equations from the literature (such as, for example, the Heston and the 3/2-model from financial engineering, (overdamped) Langevin-type equations from molecular dynamics, stochastic spatially extended FitzHugh-Nagumo systems from neurobiology, stochastic Navier-Stokes equations, Cahn-Hilliard-Cook equations) contain non-globally Lipschitz continuous nonlinearities in their drift or diffusion coefficients. A central aim of this project is to investigate regularity properties with respect to the initial values of such stochastic differential equations in a systematic way. A further goal of this project is to analyze the regularity of solutions of the deterministic Kolmogorov partial dfferential equations associated to such stochastic differential equations. Another aim of this project is to analyze weak and strong
convergence and convergence rates of numerical approximations for such stochastic differential equations.

$\bullet$ **Arnulf Jentzen**: Overcoming the curse of dimensionality: stochastic algorithms for high-dimensional partial differential equations (2019-2024)

Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. The PDEs appearing in applications are often high dimensional. Such PDEs can typically not be solved explicitly and
developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ''curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effffort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision.

$\bullet$ **Benedikt Wirth**: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization - SP: Non-smooth and non-convex optimal transport problems (2019-2022)

In recent years a strong interest has developed within mathematics in so-called "branched Transport" models, which allow to describe transportation networks as they occur in road systems, river basins, communication networks, vasculature, and many other natural and artificial contexts. As in classical optimal transport, an amount of material needs to be transported efficiently from a given initial to a final mass distribution. In branched transport, however, the transportation cost is not proportional, but subadditive in the transported mass, modelling an increased transport efficiency if mass is transported in bulk. This automatically favours transportation schemes in which the mass flux concentrates on a complicated, ramified network of one-dimensional lines. The branched transport problem is an intricate nonconvex, nonsmooth variational problem on Radon measures (in fact on normal currents) that describe the mass flux. Various different formulations were developed and analysed (including work by the PIs), however, they all all take the viewpoint of geometric measure theory, working with flat chains, probability measures on the space of Lipschitz curves, or the like. What is completely lacking is an optimization and optimal control perspective (even though some ideas of optimization shimmer through in the existing variational arguments such as regularity analysis via necessary optimality conditions or the concept of calibrations which are related to dual optimization variables). This situation is also reflected in the fact that the field of numerics for branched transport is rather underdeveloped and consists of ad hoc graph optimization methods for special cases and two-dimensional phase field approximations. We will reformulate branched transport in the framework of optimization and optimal control for Radon measures, work out this optimization viewpoint in the variational analysis of branched transport networks, and exploit the results in novel numerical approaches. The new perspective will at the same time help variational analysts, advance the understanding of nonsmooth, nonconvex optimization problems on measures, and provide numerical methods to obtain efficient transport networks.

$\bullet$ **Raimar Wulkenhaar**: Nichtperturbative Gruppenfeldtheorie durch kombinatorische Dyson-Schwinger-Gleichungen und ihre algebraische Struktur (2019-2021)

$\bullet$ **Mario Ohlberger**, **Felix Schindler**, **Tim Keil**: Localized Reduced Basis Methods for PDE-constrained Parameter Optimization (2019-2021)

This projects is concerned with model reduction for parameter optimization of nonlinear elliptic partial differential equations (PDEs). The goal is to develop a new paradigm for PDE-constrained optimization based on adaptive online enrichment. The essential idea is to design a localized version of the reduced basis (RB) method which is called Localized Reduced Basis Method (LRBM).

$\bullet$ **Mario Ohlberger**, **Felix Schindler**, **René Fritze (Milk)**, **Stephan Rave**: pyMOR - Sustainable Software for Model Order Reduction (2018-2021)

The main goal of this project is the development of infrastructures to support the sustainable development and deployment of pyMOR and related resarch software. First, we will develop and deploy a research oriented cloud service which will offer a unified development, continuous delivery and deployment workflow based on application containers. Research software will be delivered by this service for various use cases, such as continuous integration, software
demonstration, teaching or large-scale research applications. Second, we will develop guidelines for unit testing of research software in the field of scientific
computing. These guidelines will help developers to systematically write comprehensive unit tests for their software, aussuring the quality and long-term maintainability of their product.
Based on these infrastructural measures, we will improve pyMORs usability to establish our software as a universal MOR tool for various PDE-based scientific computing applications.

$\bullet$ **Benedikt Wirth**: Nonlocal Methods for Arbitrary Data Sources (2018-2022)

In NoMADS we focus on data processing and analysis techniques which can feature potentially very complex, nonlocal, relationships within the data. In this context, methodologies such as spectral clustering, graph partitioning, and convolutional neural networks have gained increasing attention in computer science and engineering within the last years, mainly from a combinatorial point of view. However, the use of nonlocal methods is often still restricted to academic pet projects. There is a large gap between the academic theories for nonlocal methods and their practical application to real-world problems. The reason these methods work so well in practice is far from fully understood.

Our aim is to bring together a strong international group of researchers from mathematics (applied and computational analysis, statistics, and optimisation), computer vision, biomedical imaging, and remote sensing, to fill the current gaps between theory and applications of nonlocal methods. We will study discrete and continuous limits of nonlocal models by means of mathematical analysis and optimisation techniques, resulting in investigations on scale-independent properties of such methods, such as imposed smoothness of these models and their stability to noisy input data, as well as the development of resolution-independent, efficient and reliable computational techniques which scale well with the size of the input data. As an overarching applied theme we focus in particular on image data arising in biology and medicine, which offers a rich playground for structured data processing and has direct impact on society, as well as discrete point clouds, which represent an ambitious target for unstructured data processing. Our long-term vision is to discover fundamental mathematical principles for the characterisation of nonlocal operators, the development of new robust and efficient algorithms, and the implementation of those in high quality software products for real-world application.

$\bullet$ **Mario Ohlberger**, **Stephan Rave**, **Marie Christin Tacke**: Modellbasierte Abschätzung der Lebensdauer von gealterten Li-Batterien für die 2nd Life Anwendung als stationärer Stromspeicher (2018-2020)