Recent concepts for controlled magneto-inertial fusion (MIF), such as magnetized liner inertial fusion (MagLIF), have suffered from magnetohydrodynamic (MHD) instabilities that lead to degradations in fusion yield. High levels of azimuthally-correlated MHD instability structures have been observed on cylindrical liner experiments without a pre-imposed axial magnetic field (Bz=0) elsewhere in the literature and are believed to be seeded from surface machining roughness. This dissertation uses highly resolved (0.5 μm and less resolution) 1D and 2D resistive magnetohydrodynamics (MHD) arbitrary-Lagrangian-Eulerian (ALE) simulations of electrical wire explosions (EWEs) and liner implosions to show that micrometer-scale surface roughness seeds the electrothermal instability (ETI), which induces early melting in pockets across the conductor and leads to millimeter-scale instability growth. The relationship between the ETI and the MRTI in liner implosions is also described in this dissertation, which shows that the traditional growth rates associated with these modes are coupled together and are not linearly independent. This dissertation also describes the preliminary implementation of a Koopman neural network architecture for learning the nonlinear dynamics of a high energy density (HED) exploding or imploding electrical conductor. / Doctor of Philosophy / Researchers have been working on controlling nuclear fusion and harnessing it as a power source since the discovery that nuclear fusion powers stars. In many of these controlled nuclear fusion concepts the aim is to heat the fuel until it forms a high-temperature plasma state of matter and then compress it to the point that the atoms are close enough and at high enough speeds that they collide fuse together. In the magnetized liner inertial fusion (MagLIF) concept these temperatures, densities, and pressures are achieved by surrounding the fusion fuel with a cylindrical piece of metal called a liner and using magnetic fields to implode the liner inward. Experiments have shown, however, that these liner implosions do not occur smoothly and that the system becomes unstable and can mix liner material into the fuel, which disrupts the fusion process. This dissertation investigates the stability of liner implosions and electrical wire explosions. In particular, this dissertation shows that surface roughness imparted on the surface of a solid fusion target by a machining process can grow into a millimeter-scale perturbation. It also describes the relationship between two common types of instabilities found in current-driven nuclear fusion: the magneto-Rayleigh-Taylor instability and the electrothermal instability. Finally, it looks at using neural networks to better understand the dynamics of electrical wire explosions.