The perceived sound of percussion instruments is significantly influenced by Viscoelastic Damping (VD), a mechanism whereby damped energy is temporarily stored and released back into the system, modifying the frequency spectrum. Performers harness this influence by choosing different materials and damping additives to shape the sound of their instruments. However, this process is still not fully understood. To address this, we apply physics-informed machine learning within a psychoacoustically grounded framework to investigate how changes in VD parameters influence the perceived sound of musical membranes. By analyzing a physics-informed dataset, we track spectral centroid differences and spectral flux across time and frequency ranges. Results show that the perceptual influence of VD strongly depends on where along the frequency spectrum the damping occurs, particularly in relation to the fundamental frequency. A temporal drift of the affected spectral region toward lower frequencies was also observed. Self-organizing maps (SOMs), initialized with physically informed features, were used to explore these complex, high-dimensional relationships. This method offers new insights into how specific damping conditions shape perceived sound and provides a foundation for more detailed analyses using alternative time–frequency representations in future research.