In this thesis, we discuss the need for enhanced sampling methods, namely to allow the simulation to sample hard to reach microstates of the system, allowing us to make better approximations of thermodynamic observables obtained from a simulation. To do this, we first introduce three enhanced sampling methods that can be applied Monte Carlo simulations with emphasis on the $2d$ Ising model. These three enhanced sampling methods are Wang-Landau sampling, a Variational Approach to Monte Carlo simulations developed by Yantao Wu and Roberto Car, and the Predictive Reweighted Autoencoded Variational Bayes for Enhanced Sampling (pRAVE) method which was developed by our group.

After introducing these three methods we then apply pRAVE to the $2d$ Ising model in the absence of an external magnetic field. Using pRAVE we explore the relative importance of the net magnetic moment to the average nearest neighbor interaction energy, and the average second nearest neighbor interaction energy as the critical temperature of the system $T_C$ is approached. We compare our results to what we should expect given that the correlation length diverges as the critical temperature is approached. We also determine the heat capacity using data from our simulation to benchmarking against Onsager's asymptotic solution near the critical temperature $T_C$. We explain how discrepancies between Onsager's results and ours can be reduced by increasing the lattice size used in our analysis. We also provide free energy plots at temperatures ranging from $T =0.9T_C$ to $T=0.999T_C$ and show that the barrier separating the two states of the $2d$ Ising model decreases as the temperature $T$ of the system is increased, in agreement with what should be observe.

After presenting our results, we discuss further work that can be done when applying pRAVE to the $2d$ Ising model as well as more complex Hamiltonians. This then paves the way for the use of pRAVE to study mechanisms for crystal nucleation as mentioned in the Conclusion of this thesis.