Differential equations in engineering and natural sciences: numerical and data-driven modeling of logistic dynamics

Abstract

Recent studies on differential equations increasingly combine classical mathemati-cal modeling with data-driven identification techniques. This trend is especially relevant in engineering and natural sciences, where real systems are often described by nonlinear dynamics, while the available data are limited or affected by noise. In such a context, a modern manuscript should not remain purely descriptive; it must demonstrate a repro-ducible computational result that quantitatively confirms the declared methodological approach.
The purpose of this paper is twofold: first, to summarize the applied role of differ-ential equations in engineering and natural sciences; second, to verify, on a reproducible control problem, the accuracy of two approaches to dynamic modeling: classical numeri-cal integration and data-driven identification of the right-hand side from limited noisy observations.
The study uses the logistic equation dN/dt = rN(1−N/K) with parameters r = 1, K = 100, N(0) = 10 on the interval t ∈ [0; 10]. The exact analytical solution is used as a benchmark. Numerical integration is performed by the explicit Euler method and the fourth-order Runge-Kutta method with a time step h = 0.5. In addition, a simplified data-driven model of the right-hand side is considered in the form f(N)=aN+bN². The co-efficients are identified by least squares from synthetic derivative data corrupted by 2% relative noise. The identified model is then reintegrated and compared with the bench-mark trajectory.
The obtained results demonstrate a substantial difference in accuracy between the two numerical schemes. For the same step size, the Euler method yields RMSE = 2.664, while RK4 gives RMSE = 0.0039. The final value N(10) is reproduced much more accu-rately by RK4. At the same time, the identified data-driven model recovers coefficients a = 1.0053 and b = -0.01006, which are close to the theoretical values 1 and -0.01. The re-constructed trajectory has RMSE ≈ 0.15, showing that even a simple parametric repre-sentation can preserve the essential nonlinear behavior and saturation near the carrying capacity.
The scientific contribution of the paper lies in methodological consistency. Instead of declaratively referring to neural differential equations without quantitative evidence, the manuscript provides a transparent comparative numerical experiment and a repro-ducible data-driven identification procedure. This removes the discrepancy between the stated goal, the applied methods, and the conclusions. The proposed framework may serve as a basis for further extensions to multidimensional systems, partial differential equations, and neural differential models with a complete training and validation pipe-line.