Rotating systems are essential components and play a critical role in many industrial sectors. Unbalance is a very common and serious fault that can cause machine downtime, unplanned maintenance, and potential damage to vital rotating machines. Accurately estimating unbalance in rotor–bearing systems is crucial for ensuring the reliable and efficient operation of machinery. This research paper presents a novel approach utilizing artificial neural networks (ANNs) to estimate the unbalance masses in a multidisk system based on simulation data from a nonlinear rotor–bearing system. Additionally, this study explores the effect of various operating parameters on oil film stability and vibration response through a combination of bifurcation diagrams, spectrum cascades, Poincare maps, and orbit and FFT plots. This study demonstrates the effectiveness of ANNs for unbalance estimation in a fast and accurate way and discusses the potential of ANNs in smart online condition monitoring systems.