# Models and Approximations

### Research Area C

Alsmeyer, Böhm, Dereich, Engwer, Friedrich, Hille, Huesmann, Jentzen, Kabluchko, Lohkamp, Löwe, Mukherjee, Ohlberger, Rave, Schedensack (until 2019), F. Schindler, Seis, Stevens, Wilking, Wirth, Wulkenhaar, Zeppieri.

In research area C, we will focus on the development and foundation of mathematical models and their approximations that are relevant in the life sciences, physics, chemistry, and engineering. We will rigorously analyse the dynamics of structures and pattern formation in deterministic and stochastic systems. In particular, we aim at understanding the interplay of macroscopic structures with their driving microscopic mechanisms and their respective topological and geometric properties. We will develop analytical and numerical tools to understand, utilise, and control geometry-driven phenomena, also touching upon dynamics and perturbations of geometries. Structural connections between different mathematical concepts will be investigated, such as between solution manifolds of parameterised PDEs and non-linear interpolation, or between different metric, variational, and multi-scale convergence concepts for geometries. In particular, we aim to characterise distinctive geometric properties of mathematical models and their respective approximations.

• ## C1. Evolution and asymptotics.

In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.

Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues.

Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees. For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities.

• ## C2. Multi-scale phenomena and macroscopic structures.

In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes.

We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flow induces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes.

We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems.

Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions.

• ## C3. Interacting particle systems and phase transitions.

The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scale randomly distributed heterogeneities and small gradient perturbations.

We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes.
We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder–Schrader axioms. Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We will also explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model.

We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interacting many-particle models of reinforced random walks. Formally, local cross-diffusion syste ms are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment.

• ## C4. Geometry-based modelling, approximation, and reduction.

In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry.

We will interpret data from different contexts (in particular measurements from the life sciences and physics, shapes from computer graphics applications, and solutions to parameterised PDEs) as elements of an underlying non-linear (infinite-dimensional) geometric space, e.g. a Riemannian manifold. This geometric structure will be exploited for the development of data processing tools. A focus will lie on variational and numerical methods for data fitting and regression via submanifolds, on singular perturbation methods, and on asymptotics and model reduction for parameterised PDEs by decomposing each solution into an element of a linear space and a Lie group element acting on it.

The geometry of spatial patterns often determines the average, effective properties of such structures, e.g. in a material, its effective material properties. Motivated by particular patterns and their defects, as observed in biological organisms or materials, we will examine their macroscopic, homogenised properties and their stability with respect to pattern perturbations. The effective, homogenised structure will again be described in geometric terms: For instance, the homogenised free energy of carbon nanotubes may depend on their bending curvature, the transport efficiency of molecules in strongly layered biological membranes of cell organelles may depend on an effective distance metric, and defects in atomic crystals can be related to specific singularities of two-dimensional surfaces.

Applications like shape optimisation or shape reconstruction problems are concerned with the identification of a geometry. Typically, there is an additional PDE constraint for which the sought geometry serves as the PDE domain. It is a challenge to efficiently approximate this geometry. We will develop concepts that quantify the efficiency of the geometry approximation in terms of the involved computational effort per desired accuracy, and we will investigate numerical schemes that can efficiently deal with complex (time-)varying PDE domains without the need for remeshing.

## 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.

$\bullet$ 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.

$\bullet$ 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 pyMORs 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)