MOAERK5: A fifth-order Runge–Kutta–based metaheuristic optimization algorithm for effective training of multilayer perceptron neural networks

Abstract

Training Multi-Layer Perceptron (MLP) neural networks is a high-dimensional, multimodal, and non-convex weight-optimization problem, and gradient-based methods such as backpropagation remain prone to slow convergence, sensitivity to weight initialization, and entrapment in local minima, motivating the search for gradient-free alternatives. This paper proposes MOAERK5, a novel population-based metaheuristic optimization algorithm derived from the six-stage explicit fifth-order Runge-Kutta (RK5) method for numerical integration. MOAERK5 uses the RK5 slope-estimation scheme to guide candidate solutions through the search space with high numerical precision, and incorporates an Enhanced Solution Quality (ESQ) operator that preserves population diversity and prevents premature convergence to local optima. In this study, the weights and biases of an MLP are encoded as the optimization variables, and the Mean Squared Error (MSE) is minimized over 500 iterations. MOAERK5 is evaluated on eight benchmark problems, five classification tasks (XOR, Balloon, Iris, Breast Cancer, Heart Disease) and three function-approximation tasks (Sigmoid, Cosine, Sine), and is compared against four established metaheuristic algorithms (HHO, GWO, WOA, SCA) over 30 independent runs. Based on Friedman and Wilcoxon signed-rank tests (α = 0.05), MOAERK5 attains the best overall Friedman rank (1.12) and statistically significant superiority over all four competitors in seven of the eight benchmarks, with the lowest error variance on the two most challenging datasets. These results indicate that coupling high-order Runge-Kutta numerical integration with population-based stochastic search offers an effective, gradient-free strategy for MLP training and other nonlinear optimization problems.