Modeling and multi-objective optimization of student engagement in virtual learning environments: a hybrid PDE-NSGA-II framework for digital music education.

Introduction: Student engagement is a key determinant of learning success in virtual learning environments, particularly in digital music education, where sustained practice and interaction are essential. This study proposes a hybrid framework that integrates partial differential equation (PDE) modeling and multi-objective optimization to analyze and improve learner engagement. Methods: Using the Predict Online Course Engagement Dataset, variables related to engagement, performance, and dropout were extracted and transformed into continuous temporal representations. A PDE-based model was developed to represent how student engagement changes over time, spreads across learning activities, and declines when participation decreases. Results: The PDE model showed stable convergence and effectively captured temporal engagement dynamics. Optimization revealed trade-offs among objectives, increasing engagement from 0.50 to 0.57 and 0.65, improving performance from 0.64 to 0.69 and 0.73, and reducing dropout risk from 0.36 to 0.30 and 0.27 under balanced and high-engagement strategies, respectively. Sensitivity analysis indicated that higher diffusion and lower decay rates improved outcomes. Discussion: The proposed PDE-NSGA-II framework provides a unified approach to adaptive learning support, offering practical guidance to enhance engagement, performance, and retention in digital music education.