In recent years, machine learned potentials (MLPs) have seen tremendous progress and rapid adoption by the materials science community. Due to their high speed and accuracy, MLPs are well suited for sampling complex potential energy surfaces (PESs) with molecular dynamics and Monte Carlo (MC) simulations. Nonetheless, many open challenges remain. Despite the outstanding performance of MLPs, it has become clear, that the builtin assumption of locality limits their applicability for systems where long-ranged effects due to charge transfer are present. But even for systems, for which local MLPs provide an adequate description, complete sampling of the PES can still be hindered by high energy barriers and advanced algorithms are needed to take full advantage of the MLPs capabilities. In this thesis, both above-mentioned challenges are addressed. In the first part, the fourth generation high-dimensional neural network potential (4G-HDNNP) is introduced. While previous generations of MLPs rely on atomic energies and charges, that only depend on the local atomic environment, the 4G-HDNNP is also able to describe long-ranged interactions caused by charge transfer effects. A charge equilibration scheme based on environment dependent electronegativities allows for the prediction of accurate atomic charges, that depend on the global state of a system, including the total charge. These charges are then not only used to compute electrostatic energies and forces, but also fed into the neural networks describing the short ranged interactions. This allows for an accurate description of changes in local bond-lengths and reactivity due to far-away changes in the electronic structure. The method’s performance is demonstrated on multiple test systems which are incorrectly described by previous methods. The second part of the thesis, focuses on the Funnel Hopping Monte Carlo (FHMC) method. FHMC introduces a new, global, MC move to directly circumvent high energy barriers that prevent complete sampling of the configuration space during MC simulations. Gaussian mixtures, fit to the Boltzmann distribution of low energy regions, are used to propose the FHMC moves without violating the detailed balance condition. FHMC therefore allows for direct sampling of the complete Boltzmann distribution without resorting to any approximate expansion of the potential energy. Anharmonic effects are therefore fully included. The method is first tested on two prototypical multi-funnel systems, namely the 38 and 75 atom Lennard-Jones clusters. We then used FHMC to study a material called methylammonium lead iodide (MaPbI3), for which we constructed a highly accurate MLP. In a recent structure search study, two non-perovskite phases of MaPbI3 were discovered, that, despite being lower in energy than the known perovskite phases, are absent in experiments. Our FHMC simulations, for which we extended the original algorithm to periodic boundary conditions, show, that above 200 K, the experimentally observed phases are thermodynamically preferred. This explains, the absence of the non-perovskite phases in experiments, since at room temperature the perovskite phases are readily obtained. The high energetic barriers then lead to kinetic trapping of the perovskite phases upon cooling.