Differential-Equation Growth Models and Machine-Learning Benchmarks for Student Enrolment in Nepalese Higher Education: A Simulation-Based Framework

Abstract

This paper develops a comparative framework for modelling student-enrolment trends in Nepalese higher education, combining three differential-equation growth models with a machine-learning benchmark. No administrative records were available: the national series (2011–2025) and the institution-level panel (60 institutions, 900 institution-years) are simulated and literature-informed. All results therefore demonstrate the behaviour of the framework and are not empirical claims about Nepal. Logistic, Gompertz and Bass diffusion models are fitted by nonlinear least squares and compared using AIC and BIC. The logistic model is selected (AIC = 58.7, against 59.8 for Gompertz and 60.7 for Bass; R2 = 0.996), with growth rate r = 0.202 per year (95% bootstrap CI 0.188–0.216) and carrying capacity K = 519.8 thousand (95% CI 504.6–539.0). Linearisation shows that K is a globally stable equilibrium in all three models, and a log-transform argument establishes global stability for the Gompertz model. Explicit Euler and fourth-order Runge–Kutta integration are benchmarked against a fine-step reference, with fitted log–log slopes of 1.01 and 4.02. A 1,500-draw residual bootstrap propagated into a Monte Carlo forecast gives a 95% envelope of 501–529 thousand students for 2035. In the institution-level benchmark, linear regression attains the lowest cross-validated RMSE (35.7 thousand, against 39.2 for random forest and gradient boosting) and the highest test R2 (0.687), a result that follows from the near-linear design of the simulated data. The Python framework, with MATLAB equivalents, can be re-run directly on genuine enrolment records when they become available