A scientific method for evaluating the design of interfaces is proposed to address the unique characteristics and user needs of infrequent-contact public service interfaces. This method is significant for enhancing service efficiency and promoting the sustainable development of public services. Current interface evaluation methods are limited in scope and often fail to meet actual user needs. To address this, this study focuses on virtual museums, examining users’ aesthetic psychology and cognitive behavior in terms of layout aesthetics and visual cognitive characteristics, aiming to explore the relationship between the two. Interface layout aesthetic values and user visual cognitive measurements were obtained by using computational aesthetics methods and eye-tracking experiments. These served as input data for a new model. An improved gray H-convex correlation model utilizing the ICRITIC method is proposed to examine the mapping relationship between interface layout aesthetics and visual cognitive features. The results demonstrate that our new model achieves over 90% accuracy, outperforming existing models. For virtual museum interfaces, symmetry and dominance significantly influence user visual cognition, with the most notable correlations found between density and gaze shift frequency, simplicity and mean pupil diameter, and order and gaze shift frequency. Additionally, fixation duration, fixation count, and mean pupil diameter were inversely correlated with interface layout aesthetics, whereas gaze shift frequency and gaze time percentage were positively correlated.