The rising demand for rechargeable batteries, driven by renewable energy integration, highlights the limitations of conventional Li-ion batteries. Alternative battery technologies promise superior energy storage but struggle with dendrite formation during electrodeposition, posing efficiency and safety challenges. Conventional numerical methods like the finite element method (FEM) used to analyze dendrites require extremely detailed meshes, leading to high computational costs. Recently, physics-informed neural networks (PINNs) have in some cases shown to outclass these conventional numerical methods. In this thesis, PINNs are employed to a variety of phase-field models. Given that PINNs are mesh-free solvers for partial differential equations (PDEs), they hold promise in this context, aiming to rival FEM while significantly reducing the computational cost. For a simple binary phase separation, PINNs agree qualitatively with FEM, both for smooth interfaces, and sharp interfaces, where PINNs are known to struggle. When studying solidification problems, PINNs can learn the correct dynamics when solving in one spatial dimension, albeit at a slightly slower timescale than FEM, but completely break down in two spatial dimensions. A possible reason for this is a significant jump in the complexity of the PDE, accompanied by a low resolution at the interface between the phases, which drives the dynamics. Lastly, an electrochemical phase-field model is studied. Here, PINNs encounter challenges in computing solutions across large domains and long timescales, highlighting their difficulty in tackling multiscale problems. In all of the studied cases, PINNs were unable to compete with FEM in terms of accuracy and computational cost.