In the doctoral research projects we will *combine in novel ways analytical,
numerical, and modeling aspects* related to energy/entropy functionals and their
dissipation mechanisms.

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## Materials Science

*Michael Herty, Maria G. Westdickenberg*

It is well-known that large deviation theory provides sharp exponential bounds on rare events in the small noise limit. Entropic effects are less well understood. In an Allen-Cahn type equation with an asymmetric two-well potential, the broad well is preferred by the stochastic process over the narrow one even though the local minimizers of the potential have the same energy. As a consequence, the equilibrium distribution of transition layers is no longer uniform and pairs of transition layers tend to stay close together. The entropic effect of the large system size, on the other hand, is that transition layers tend to spread. The dynamics of stochastically perturbed PDEs in the presence of entropic effects can be formally approximated using a generalization to infinite dimensions of a theorem of Katzenberger. Some of the effects observed there are ones that we would like to look at more closely numerically and analytically. This project combines the development of efficient numerical methods for stochastic dynamics with the analysis of coarsening and nucleation events. Part of the work extends the notion of asymptotic preserving schemes to this context in order to correctly resolve dynamics independent of the numerical discretization. The topic of asymptotic preserving schemes has been studied extensively, e.g., in the context of kinetic equations and their hyperbolic limit. Also, in the context of continuous optimization, studies of limiting schemes and limiting optimization problems have been investigated recently in simpler systems.

*Umberto L. Hryniewicz, Maria G. Westdickenberg*

For an initial curve asymptotically close to a circle, Xinfu Chen (1993)
established global existence of the Mullins-Sekerka evolution on a bounded two
dimensional domain. His method relies on linearization and a Schauder
contraction argument. Joint work of ours with Chugreeva and Otto (2019) applied
a more *nonlinear*, energy-based argument to establish sharp convergence rates
to the half-plane for initial data order one away from that state. Extending
such a framework to functions order one away from a circle requires new
estimates and/but could yield a much stronger global existence result as well as
convergence rates. The underlying theme/tool is the use of our
Distance-Energy-Dissipation method developed in Otto-Westdickenberg (2014),
which extends an observation of Brézis to the nonconvex setting.

*Christof Melcher*

Topological defects and solitons are ubiquitous in condensed matter systems such as ferromagnetism, liquid crystals, and Ginzburg-Landau models of superconductivity or Bose-Einstein condensation. An attractive and widely open research field is concerned with understanding processes of nucleation and annihilation within the framework of corresponding dissipative evolution equations such as Landau-Lifshitz or Ginzburg-Landau equations. While the spontaneous formation of singularities is fairly well understood, not much is known about how to initiate or trigger such a singular process in a controlled and possibly optimal manner—which is a key issue in applications, e.g., in future information technologies. An important question along the way is to understand the underlying energy landscape and energy barriers for the tunnelling between topological sectors.

*Michael Herty, Heiko von der Mosel*

We investigate the elastic behavior of knotted loops of springy wire. The analytical and numerical considerations are based on energy minimization principles, for which the minization iteration along the energy landscape can be interpreted as an evolution in time. The landscape in given knot classes is described in the simplest elastic setting by the classic bending energy plus a small multiple of a repulsive potential $\mathcal{R}$ that penalizes self-intersection. The existence of energy minimizing configurations of the resulting total energy in every given knot class is settled for various choices of the potential $\mathcal{R}$. Much less is known for the limiting elastic knots where the strength of $\mathcal{R}$ is sent to zero, or the situation with elastic knots with twists. Our aim is to derive new energy decay rates and an improved convergence of the energy minimizers towards the respective elastic knots in more general knot and link classes. This will require the careful construction of suitable comparison curves. We also want to design numerical schemes that iteratively minimize the knot energy and to gain insight for knot classes for which no analytical results are available right now.

## Hyperbolic PDEs

*Michael Herty, Siegfried Müller*

In hyperbolic conservation laws, uncertainties may be introduced, for instance, by initial or boundary data as well as fluxes. In order to capture the effect of such uncertainties on the solution we are interested in quantities like expectation and variance of solutions. The resulting equations pose serious challenges in the analysis as well as in the numerical resolution. While there has been tremendous progress in the field of stochastic elliptic equations similar results for hyperbolic are still at large. Within the research project we want to examine efficient and reliable numerical schemes for stochastic hyperbolic conservation laws. Here we plan to focus on intrusive methods based on recent results of the applicants on entropy-entropy flux pairs for one-dimensional stochastic hyperbolic equations. In contrast to non-intrusive methods like Monte-Carlo methods, where many deterministic problems are solved and averaged over the stochastic domain, intrusive methods modify the underlying equations, e.g., by stochastic Galerkin projection. These methods are based on a generalized polynomial chaos expansion of the solution $u$ in the space of all square integrable random variables with respect to an orthogonal polynomial basis.

*Michael Herty, Siegfried Müller, Michael Westdickenberg*

In recent years there has been tremendous progress in the mathematical theory of transport and continuity equations, starting with the seminal work by DiPerna-Lions (1989) and Ambrosio (2004) who generalized the classical theory to the case of velocity fields with Sobolev and $\mathrm{BV}$-regularity, respectively. The main difficulty stems from the fact if the velocity field is not Lipschitz continuous, then solutions of the corresponding ordinary differential equations may no longer be unique. The Lagrangian point of view plays an important role in these developments, as formalized in Ambrosio’s regular Lagrangian flow and the interpretation of solutions as superpositions of integral curves of the underlying ordinary differential equation. Bianchini-Bonicatto (2017) proved the stability of the regular Lagrangian flows, known as Bressan’s conjecture, for convergent sequences of $\mathrm{BV}$-regular velocity fields. The existence of a suitable flow enables us to split transport equations into a geometry part, captured by the flow, and the transport part, which significantly simplifies since the evolution only needs to be studied along the characteristics. This research project aims at developing novel numerical schemes for transport equations based on this insight, by connecting these theoretical results with recent breakthroughs in discontinuous Petrov-Galerkin methods for transport equations by Broersen-Dahmen-Stevenson (2015) and others.

*Michael Westdickenberg*

Well-posedness of the compressible Euler equations of gas dynamics is still on
open problem in more than one space dimension. Recent work by De
Lellis-Székelyhidi (2009) and others showed that for suitable initial data
*infinitely many weak solutions* can be constructed using techniques inspired by
convex integration. It is still unclear whether a suitable entropy condition can
restore uniqueness and result in the well-posedness of the problem. In this
research project we will investigate an approach based on a suitable **selection
procedure**: Within the class of dissipative solutions, a solution concept akin
to measure-valued solutions, we seek to find the one that minimizes the
acceleration (or the total energy) at a.e. instance of time. The existence of
such a minimizing dissipative solution has been established recently. In a
sense, this solution is as close to a weak solution as possible. We will further
investigate the properties of such extremal solutions and explore possible
connections with the approach by De Lellis-Skèkelyhidi.

*Michael Westdickenberg*

It is well-known for ordinary differential equations of the form $\dot{x} =
\mathbf{b}(t,x)$, that if the velocity field $\mathbf{b}$ is not Lipschitz
continuous, then solutions of the corresponding initial value problem may not be
unique. As a consequence, it is not clear how to define a flow: which integral
curves should one pick? This difficulty is mirrored in the question whether the
transport or continuity equations involving the same velocity field $\mathbf{b}$
may be ill-posed. A lot of effort has been devoted in the last decades to
identify assumptions on $\mathbf{b}$ that would still allow to define a suitable
version of flow, such as Ambrosio’s regular Lagrangian flow. These assumptions
typically come in the form of Sobolev- or $\mathrm{BV}$-regularity of
$\mathbf{b}$. In this research project we will investigate a different approach,
which we introduced recently: Instead of trying to force *essential uniqueness*
of solutions of the underlying ordinary differential equation, we try to
**select** integral curves for a.e. starting point in such a way that the
resulting family of curves form a flow. This approach is motivated by Krylov’s
Markov selection in the theory of stochastic processes and is possible under
fairly weak regularity assumptions on the velocity. Once a flow is selected, the
corresponding transport problem is well-posed.

*Michael Herty, Manuel Torrilhon*

The closure problem of moment equations of the Boltzmann equation requires the inference of a distribution of the particle velocities from a finite set of moments. This reconstruction task is underdetermined and requires additional conditions on the distribution, like the minimization of a certain functional. Classic closures use a weighted $L^2$-norm (Grad-Hermite closure) or the entropy (minimum-entropy closure) as functional. In this project we want to explore a closure that minimizes the Wasserstein distance of the distribution to the local equilibrium (Maxwell) distribution under the constraint of given moments. Through the interpretation as optimal transport this closure would also have a physical motivation of choosing a distribution that requires minimal transport costs of the particle from the equilibrium state into the current state. In some sense this also resonates with the entropy approach where the most likely distribution is chosen. The optimal transport map would have implications on the resulting structure of the moment equations, like hyperbolicity and H-theorem. Numerically, this closure would be implemented in a discretized distribution setting and tested on scenarios like a Riemann problem or stationary shock profile.

## Optimal Transport

*Siegfried Müller, Michael Westdickenberg*

Optimal transport is concerned with the problem of moving one probability
measure into another one while minimizing the total cost of the transport. If
the cost is given by the squared distance, then the optimal transport plan is
contained in a cyclically monotone set, i.e., the subdifferential of a convex
function. With sufficient regularity, the condition expressing compatitibility
of the initial and final measures then amounts to a Monge-Ampère equation for
the transport map, which is a *nonlinear elliptic equation*. It is indeed
possible to solve the optimal transport problem by solving the Monge-Ampère
equation. In the available literature the main focus is on solving a suitable
discretization of this nonlinear equation using some sophisticated variant of
Newton’s method, for example. What has not been fully addressed, however, is the
question of **monotonicity** of the transport map. This will be investigated in
this research project. Similar issues arise in the design of mirrors/lenses that
have the property that light from a point source is reflected/refracted so that
the resulting intensity matches a prescribed light distribution. The goal of
this project is to develop efficient numerical methods for these and related
problems.

*Heiko von der Mosel, Michael Westdickenberg*

The theory of causal Fermion systems has been proposed as a physical model that can unify weak, strong, and electromagnetic forces with gravity. Unlike other physical theories that describe objects on a pre-existing manifold, the theory of causal Fermion systems aims to derive space-time itself together with the physical objects and structures that inhabit it. There are very interesting mathematical problems originating from this theory, which is formulated in terms of causal variational principles that generalize the causal action principle. In a simplified setting, this boils down to minimizing a non-negative, lower semicontinuous action functional, which is given as a double integral of Borel measures against an explicitly known, concave integrand. The support of the minimizing measure is identified with space-time in the theory of causal Fermion systems. It is expected to be a lower-dimensional or even a discrete subset of the underlying topological manifold. In recent work it was proved that the support is contained in a finite number of analytic curves, which intersect at a finite number of points. From this, an upper bound on the dimension of the support could be derived. In this research project, we will advance this line of research by exploiting the similarity of the causal variational principles with optimal transport problems with concave cost functions, for which the optimal transport plans can be characterized geometrically.

## Machine Learning

*Michael Herty, Manuel Torrilhon*

In the context of large–scale machine learning problems modern optimization methods are based on the evolution of a particle system. Here, the particles evolve in a possibly highly non-trivial landscape towards a (local) minimum of a suitable energy functional. Currently most of the proposed methods are heuristics and suitable convergence results are still incomplete. In this project we propose to analyze and exploit possible energy and entropy dissipation properties of corresponding mean-field equations to gain insight on convergence rates and stability of the discovered local minima. A first preliminary example has been studied by Carrillo et al. (2018) where a set of particles evolves within an energy landscape according to a Kuramoto-Vicsek type model. The particles are instantaneously driven towards a weighted average of the particle positions which represents an approximation of the function minimum. An additional stochastic component is here required to ensure robustness of the method and to obtain convergence estimates for the corresponding mean-field equation. In this project, we plan to extend the concept of particle methods in several directions. This will be done by examining a variety of dynamic models also posed on suitable, not necessarily metric, spaces that will be studied both analytically and numerically. Moreover, we plan to extend the current results on mean-field level to Boltzmann-type equations that allow for the existence of an analogue to the classical H-Theorem. Large time behavior of the system will also be studied by suitable mean-field descriptions as well as constrained dynamics. Ideally, decay rates for suitable energy and entropy functionals in Wasserstein distance will be obtained. Furthermore, the stability of the local minima will be investigated. Finally, the theoretical results should be accompanied by fast and efficient, possibly higher-order discretizations to obtain feasible methods. Those will also be used to solve numerically general constrained optimization problems.

*Michael Westdickenberg*

Gradient flows have been extensively studied in the context of Hilbert spaces or
Riemannian manifold for a very long time. In recent decades, gradient flows have
also been considered on spaces of probability measures since various parabolic
partial differential equations can be (formally) interpreted as a gradient flows
with respect to a suitable metric structure, induced for example by the
Wasserstein distance. This has led—among other things—to the development of
the theory of *curves of maximal slopes*, which generalizes gradient flows to
rather general metric spaces. In this research project, we will investigate
gradient flows on space of matrices where completely new features appear due to
the non-commutativity of the matrix multiplication. There are two important
applications for this research: First, gradient flows on matrix spaces occur
naturally in the context of dissipative quantum systems, as observed by
Carlen-Mas (2014). Second, deep learning of neural networks often amounts to a
minimization over a space of weights, which are matrices. These minimization
problems can be solved using gradient descent methods, which can be understood
as time discretized versions of gradient flows. Since the objective functions
are typically non-convex, there are many open problems related to the
convergence of such descents/flows and the characterization of their long-time
limits.

*Maria G. Westdickenberg, Michael Westdickenberg*

Mathematically, deep learning of artificial neural networks can be understood as the attempt to find a function that maps given data to expected outcomes. A prototypical example is to learn, from an image of a hand-written letter, which letter the image actually represents. The ansatz space for this approximation problem consists of functions that are compositions of affine maps and pointwise activation functions (such as the rectified linear unit ReLU), over several layers. Deep learning then amounts to optimizing the matrix weights and other parameters that specify the affine functions in each layer. Typically they are defined on high-dimensional vector spaces, where the dimension can be the number of pixels of an image, for example. In this project, we will be interested in the limits of either (1) infinite width: the dimension of the vector spaces is sent to infinity, in which case matrices are replaced by functions/measures; or (2) inifinite depth, i.e., the number of layer becomes unbounded. In either of these cases, the discrete nature of artificial neural networks gives way to continuously defined structures whose properties can be studied using ordinary or partial differential equations. In the case of infinite depth, for example, deep learning can sometimes be rephrased as an optimal control problem; for infinitely wide networks, one may obtain a description in terms of stochastic processes and Fokker-Planck-equations.

## Symplectic Geometry

*Umberto L. Hryniewicz, Michael Westdickenberg*

The notion of entropy in dynamical systems (topological or measure theoretic) goes back to the work of Kolmogorov, Bowen, and Adler, Konheim, and McAndrew. Topological entropy is a conjugacy invariant that quantifies chaotic behaviour inside the system. During the second half of the twentieth century entropy has become a central concept in dynamical systems theory and remarkable results have been proved. For instance, Katok showed that (in low dimensions) a smooth system with positive topological entropy admits compact invariant sets where the dynamics is conjugated to that of a topological Markov chain with exponential growth of periodic orbits. This research project is about detecting entropy in low dimensional Hamiltonian systems. We wish to explore to what extent a given collection of periodic orbits of a smooth area-preserving diffeomorphism on a surface, or of a Hamiltonian flow on a three-dimensional energy level, forces positivity of the entropy. Notice that there are situations where the ambient phase space has a complicated structure that forces positive entropy for all Hamiltonian systems defined on it. This will, however, not be the focus of this research project. Instead our intention is to study simple phase spaces that do admit zero entropy systems, and then to investigate patterns that will force the entropy to be positive.

*Umberto L. Hryniewicz, Heiko von der Mosel*

The aim of this research project is to investigate systematically the relations between the $\Gamma$-convergence of energy functionals when homogenized in configuration variables, and the corresponding limiting processes on the dual Hamiltonian side. Possible applications include regularity problems for obstacle problems for elastic rods where a varying parameter in the energy functional could penalize extensibility of the minimizing rod in favour of an inextensible limit regime. Sets of vanishing velocity of the extensible rod’s centerline, which cause regularity problems, will then be reduced during the limiting process. More generally, the dependence of holonomic constraints in dynamical systems on varying parameters in the energy density is reflected on the Hamiltonian side, and this needs to be examined rigorously in the various multiplier formulations suggested, e.g., by Gonzalez, Maddocks, and Pego. On the symplectic side, the Lagrangian point of view might be a way to better understand the topology of the group of Hamiltonian diffeomorphisms.