Open Access

Dynamical systems consist of ordinary and partial differential equations and are among the most prominent approaches to model physical processes. They describe the evolution of the system in terms of the current system state and external control inputs. To improve model accuracy, it can be beneficial to explicitly include properties such as energy conservation or energy dissipation, which are usually present in the real-world phenomena, in the mathematical problem description. Ideally, these properties should be kept in mind whenever one interacts with the model. The focus of this thesis is on two kinds of interactions: the discretization of such models, and their control. Discretization techniques are necessary whenever the system evolution is to be approximated using computational methods. Similarly fundamental is the numerical realization of control inputs that lead to desired outcomes. Both aspects require special attention to retain an energy-based perspective. For this, the first step is usually to encode the energy properties in the algebraic description of the model. This description must strike a balance between general applicability and its corresponding benefits. The next step is to leverage the algebraic description in further analysis. While the field is well-developed for linear systems, the nonlinear case often poses additional difficulties. First, there seems to be no clear consensus about what model class to use to describe nonlinear physical phenomena. Second, many discretization methods that preserve the energy-based viewpoint in the linear case are not trivial to generalize to nonlinear systems. Third, studying the behavior of energy-optimal controls and finding control laws that can be realized via energy-based models is usually more difficult. In this thesis, these points are addressed. We give an overview of energy-based model classes used for nonlinear phenomena in both finite and infinite dimensions and investigate their relationship. Moreover, using a modified Petrov--Galerkin method and discrete gradients, we present multiple structure-preserving discretization schemes. Furthermore, we show that similar to the linear case, energy-optimal controls steer the associated trajectories to the submanifold of the state space where no dissipation is present. Finally, we combine an optimal feedback law characterized by the Hamilton--Jacobi--Bellman equation with output feedback to state an energy-based feedback controller. Our theoretical results are illustrated using numerical experiments.

A T-curve of degree d is given by a regular unimodular triangulation of d⋅Δ2 together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves.

We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach.

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B∈R≥0 is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4ℓ−1)χ,(2^(ℓ+2))/(2^ℓ−1))-approximation for every fixed ℓ∈N.