Machine Learning in Inverse Problems - Learning Regularisation Functionals and Operator Corrections

Abstract: This thesis deals with the analysis and numerical simulation of anisotropic nonlinear partial differential equations (PDEs) and dynamical systems in biology. It is divided into two parts, motivated by the simulation of fingerprint patterns and the modelling of biological transport networks. The first part of this thesis deals with a class of interacting particle models with anisotropic repulsive-attractive interaction forces and their continuum counterpart. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing interacting particle models, the forces are isotropic and the continuum limits of these particle models are given by nonlocal aggregation equations with radially symmetric potentials. The central novelty in the models we consider is an anisotropy induced by an underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but also renders their analysis significantly harder compared to their isotropic counterparts. We discuss the role of anisotropic interaction, study the steady states and present a stability analysis of line patterns. We also show numerical results for the simulation of fingerprints, based on discrete and continuum modelling approaches. The second part of this thesis focuses on a new dynamic modeling approach on a graph for biological transportation networks which are ubiquitous in living systems such as leaf venation in plants, blood circulatory systems, and neural networks. We study the existence of solutions to this model and propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. We also show the global existence of weak solutions of the macroscopic gradient flow. Results of numerical simulations of the discrete gradient flow illustrate the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters. Based on this model we propose an adapted model in the cellular context for leaf venation, investigate the model analytically and show numerically that it can produce branching vein patterns.