Neural Network Approaches to High-Order Extensions of Simpson-Type Inequalities

M12-1 2nd KOCAELI SCIENCE CONGRESS (KOSC-2025) Kocaeli University, Faculty of Arts and Sciences November 19-21, 2025, İzmit, Kocaeli, Türkiye Neural Network Approaches to High-Order Extensions of Simpson-Type Inequalities Canmert DEMİR1, Samet ERDEN2 1Department of Computer Science, Istanbul Rumeli University, İstanbul, Turkey 2Department of Mathematics, Bartin University, Bartın, Turkey Corresponding author: [email protected] ORCID IDs: First Author: 0009-0008-5538-6242 Second Author: 0000-0001-8430-7533 DOI : 10.5281/zenodo.18017355 Abstract We investigate the fundamental inequalities approximation, which Erden et al. introduced in literature, is stated in this work. For the previously mentioned classes of functions, the Simpson-type quadrature rules derived from these inequalities are then investigated. The Levenberg–Marquardt (LM) technique is used to train an artificial neural network (ANN) to approximate numerical integration results based on Simpson-type inequalities. It aims to increase the computational accuracy of higher-order derivative-based computations while lowering training error metrics such as Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE). The results demonstrate that the LM-trained ANN effectively models nonlinear mappings inside Simpsontype inequality frameworks, providing a workable numerical alternative to traditional analytical integration methods. Because it effectively captures the basic relationships between input parameters and their associated integral approximations, this adaptive network may generalize across a range of functional behaviors that are often challenging for traditional approaches to depict. Compared to existing numerical and analytical methods in the literature, this computational design shows improved precision, smoother convergence dynamics, and increased robustness to local minima during the optimization process. Furthermore, the LM-based training approach provides a flexible and scalable approach to extend the applicability of Simpson-type inequalities to more complex domains. Overall, the proposed Levenberg–Marquardt ANN provides a data-driven, flexible, and highly accurate alternative to traditional numerical integration methods. It effectively models complex nonlinear behaviors and extends Simpson-type inequalities into a computational learning framework, offering new potential for hybrid analytical–neural approaches in high-precision quadrature analysis. Keywords: Simpson-type inequalities, Quadrature rules, Artificial Neural Network (ANN), Levenberg–Marquardt algorithm, Numerical integration, Error analysis, Data-driven modeling. 1. Introduction and Motivation The Simpson's rule, developed by Thomas Simpson (1710–1761), is an important numerical integration method for the approximate calculation of definite integrals. Since then, many scholars have expanded and generalized Simpson-type inequalities, and one of the most notable results frequently cited in the literature within this framework is Simpson’s inequality, first introduced in [3]. It can be stated as follows. M12-2 2nd Kocaeli ScienceCongress, November 19-21, 2025 (1.1)󰈅 𝑓(𝑥)𝑑𝑥−𝑏−𝑎 3󰇩𝑓(𝑎)+𝑓(𝑏) 2+2𝑓𝑎+𝑏 2󰇪  󰈅 ≤ 1 2880𝑓()(𝑏−𝑎) where the mapping 𝑓 :[𝑎,𝑏]→𝑅 is assumed to be four times continuously differentiable on the interval (𝑎,𝑏) and for the fourth derivative to be bounded on (𝑎,𝑏) , that is, 𝑓() := 𝑠𝑢𝑝 ∈(,) 𝑓()(𝑥)<∞. Simpson’s inequality, which bounds the error of Simpson’s rule using the fourth derivative of the integrand, is a fundamental result in numerical analysis. Dragomir et al. [3] advanced this inequality by proposing new applications, tight ererror bounds, and extensions to various function classes, including bounded, Lipschitz, and absolutely continuous functions with in Lebesgue spaces. Their contributions laid the ground work for subsequent generalizations of Simpson-type inequalities under different smoothness and convexity assumptions. Liu [9] further extended the concept by deriving a Simpson-type inequality applicable to functions that are continuously differentiable n times. 𝐹𝑜𝑟 𝑛= 1,2,3,4 , these results recover the classical or strengthened forms of Simpson’s inequality, while for general n they yield perturbed variants. Recent research has introduced new integral identities and generalized notions of convexity, thereby broadening the scope of Simpson-type inequalities. Alomari et al. [1] formulated new inequalities for s-convex and convex functions, established corresponding error bounds, and demonstrated their utility in mean value computations and numerical integration. Similarly, Liu [12] extended Simpson-type inequalities to n-times continuously differentiable functions, providing generalized error estimates that improve the reliability of numerical approximations. Composite Simpson’s formulas have been applied to convex functions [16], highlighting the significance of continuity and higher-order differentiability for error minimization in both theoretical and applied contexts. Fink [10] generalized Ostrowski’s inequality to functions differentiable of arbitrary order, while Anastassiou [2] developed related inequalities for functions in 𝑓∈𝐶([𝑎,𝑏]). Other studies investigated the relationship between exact integral values and their approximations obtained via quadratic formulas derived from higher-order derivative inequalities. Building on this framework, Kashif et al. [14] proposed a three-step kernel for n-times differentiable functions, which Qayyum et al. [15] later expanded into a five-step kernel to enhance the efficiency of quadrature rules. In [4], a five-section quadratic kernel was utilized to derive novel integral equalities for functions with absolutely continuous higher-order derivatives, along with perturbed Ostrowski-type inequalities for bounded and bounded-variation functions. Erden et al. [6,8] extended these methods, developing integral inequalities for higher-order differentiable functions, offering error estimates for numerical approximations, and analyzing Simpson-type quadrature formulas. These studies not only advance the theoretical understanding of Simpson-type inequalities but also improve error management in numerical integration and approximation. Erden et al. [7] refined Simpson-type inequalities for higher-order absolutely continuous functions, providing tighter bounds and enhanced error estimates for practical computation. More recently, artificial intelligence techniques particularly neural networks and deep learning models [4, 11] have been successfully applied to approximate solutions of differential equations and fractional integral inequalities, achieving high accuracy where conventional methods may be insufficient. Additional studies [5,9] have introduced new fractional inequalities without assuming convexity, validated via ANN models, demonstrating both theoretical significance and practical applicability. 2. Material and Method 2.1. Comparison of Quadrature Rules, ANN, and Simpson Integral Inequality Using inequalities from Erden et al. [7,9], this section investigates Simpson-type quadrature rules for functions that belong to 𝐿 spaces or have bounded higher-order derivatives. Two methods are compared: (1) symbolic analysis utilizing derivative-based integral techniques, and (2) numerical M12-3 2nd Kocaeli ScienceCongress, November 19-21, 2025 evaluation using an ANN model and the Levenberg–Marquardt algorithm. With quadrature criteria provided in [9], both approaches deal with the theoretical generalization and numerical verification of Simpson-type inequalities. (2.1) 𝑄,(𝑓,𝐼) :=󰇫 (−1) 2⋅(𝑘+1)!󰇩𝑏−𝑎 2+(−1)𝑏−𝑎 6󰇪   ×𝑓()𝑎+2𝑏 3+(−1)𝑓()2𝑎+𝑏 3 +  1 (𝑘+1)!   𝑏−𝑎 6𝑓()(𝑎)+(−1)𝑓()(𝑏) By comparing symbolic expressions to exact integrals, the ANN model is trained using data from 𝑄, to evaluate the behavior and accuracy of Simpson-type approximations. Strong convergence is ensured via the Levenberg–Marquardt algorithm, which produces approximate integrals for sample functions and goes through normalization, training, and testing. The symbolic approach, on the other hand, provides a theoretical foundation for error analysis and clearly identifies sources of mistake by analytically computing derivatives and integrals. Using integration limits, iteration index, and boundary values as inputs, an artificial neural network (ANN) was created to forecast Simpson-type inequalities. Using mapminmax normalization and an 75/15/15 train–validation–test split, the feed forward network—which has a 10-neuron hidden layer (logsig activation) and linear output—was trained using the Levenberg–Marquardt algorithm, successfully capturing nonlinear relationships and offering trust worthy predictions when analytical computation (2.2) 𝑀𝑆𝐸= 1 𝑛(𝑦−𝑦 )   where n is the total number of data points, 𝑦 values are real values, and 𝑦  values are projected values. Better learning and generalization skills are shown by a reduced Mean Squared Error (MSE) score, which also demonstrates how well the model's predicted outputs match the real target values. With a low mean square error (MSE), excellent accuracy, and reliable performance on untested data, the neural network successfully captured the nonlinear relationships between inputs and outputs. By examining the relationship between prediction and actual values, the regression coefficient 𝑅 was used to assess the network's efficacy even more. The regression equation's expression is: (2.3) 𝑅= ∑ (𝑦− 𝑦)𝑦 −𝑦    ∑∑ (𝑦− 𝑦)   ∑ (𝑦 −𝑦)     where 𝑦 and 𝑦  represent the mean values of the expected and actual outputs, respectively. 𝑅 has a range of −1 to 1, where 𝑅=1 indicates a very good prediction, 𝑅=0 indicates no correlation, and 𝑅=−1 indicates an ideal negative correlation. M12 - 4 2nd Kocaeli ScienceCongress, November 19 - 21, 2025 Fig2 Comparison of the True, Reference [9], and ANN-Based Integral Surfaces 𝑓 ( 𝑥 ) = 𝑥𝑙𝑜𝑔𝑥 Fig1 ANN Architecture for Simpson-Type Integral Approximations Three different integral surfaces that were discovered for the function 𝑓(𝑥) = 𝑥𝑙𝑜𝑔𝑥 are now compared. First, the precise integral surface derived from the function's analytical solution serves as the reference. After obtaining the approximation surface using the method recommended in [9], the artificial neural network (ANN–based approximation) is evaluated. The suggested ANN methodology produces results that are strikingly close to the genuine values and the method in [9], as the comparative graph of these three surfaces demonstrates. This outcome shows how effectively the model predicts and logarithm function behavior. At this point, the Levenberg–Marquardt algorithm-trained neural network model's performance is assessed from four angles. First, the blue circles (actual values) and red squares (predicted values) on the graph nearly entirely overlap, showing that the model accurately fits the training data. M12 - 5 2nd Kocaeli ScienceCongress, November 19 - 21, 2025 Fig3ContourMapComparison of Surfaces In summary, the model retain slow and well-controlled prediction errors, was successfully trained, and shows no indications of over fitting. The reference surface produced from analytically calculated integral values is represented by the Exact Integral Surface, which exhibits smooth, continuous variation along parameters 𝑎 and 𝑏. With a very comparable curvature and color distribution, the Simpson-type 𝑄 , approximation surface substantially resembles this reference, suggesting that the 𝑄 , approach accurately captures the integral values. The Exact and 𝑄 , surfaces, on the other hand, are very different from the Artificial Neural Network (ANN) anticipated surface. It has a nearly one-dimensional structure, changing only significantly along parameter 𝑎 and responding very little to changes in 𝑏. This pattern implies that the ANN did not fully understand the target function's true twodimensional dependence, most likely as a result of constraints in the training dataset or the network's capability. Related graph given below in Fig. 2. The contour maps of the Exact and 𝑄 , surfaces exhibit a high degree of similarity, with nearly identical contour orientations, densities, and color distributions. This resemblance once again demonstrates that the 𝑄 , method closely follows the true integral values with strong accuracy. In contrast, the ANN-predicted contour map displays vertically aligned bands, indicating that the model’s output varies primarily with respect to parameter 𝑎 while showing minimal sensitivity to changes in 𝑏. This pattern confirms that the ANN failed to capture the two-dimensional structure of the underlying integral function. Overall, the results show that the ANN model did not successfully learn the full bivariate relationships present in the target function Fig. 3. M12-6 2nd Kocaeli ScienceCongress, November 19-21, 2025 Fig4 Train, Test, Validation Perfomance Metric Fig5 Performance Metric for Training, Testing, and Validation The training, validation, and test errors all exhibit a typically declining trend, according to the training error graph (MSE vs. epoch). The seventh epoch yields the best validation performance, with a value of 7.26 × 10. Nevertheless, the model's projected surfaces do not match the actual functional organization, even with the minimal numerical error. This implies that even while the ANN achieves a low MSE, its learnt representation differs from the target function's real behavior. The M12-7 2nd Kocaeli ScienceCongress, November 19-21, 2025 model's inability to fully capture the surface's complexity is probably caused by an unbalanced training set or an inadequately expressive network architecture. In conclusion, the ANN shows structurally erroneous learning and is unable to represent the function's dependence on parameter (b) despite the low MSE in Fig 4 and Fig 5. Fig6 Error Histogram The prediction behavior related to the Simpson-type inequality estimation is examined using the error histogram of the ANN model. The figure shows the distribution of residual errors between the predicted and true integral values for the training, validation, and test subsets. One may determine whether the network's predictions show random fluctuations or systematic bias by looking at the density and symmetry of these residuals. Furthermore, a numerical indicator of the model's depend ability is provided by the standard deviation of the residuals, or 𝜎 which quantifies the dispersion of prediction mistakes. A greater 𝜎 denotes greater unpredict ability and decreased stability, whereas a lower 𝜎 shows that the ANN forecasts consistently stay near the true values. The following formula is used to calculate the standard deviation, which describes the characterizes the overall spread of the error distribution. M12-8 2nd Kocaeli ScienceCongress, November 19-21, 2025 Fig7 Absolute Error Desricbes in [9] Surfaces 𝑄, and ANN The method 𝑄,, approximates the exact surface with a remarkably small error; in terms of both surface geometry and contour structures, it exhibits an agreement with the analytical solution that is practically indistinguishable. In contrast, while the ANN model successfully mimics the true surface in certain regions of the 𝑎−𝑏 domain, it fails to achieve the same level of accuracy as the approach proposed by [9] in other regions. A particularly notable weakness is observed in the model’s inability to learn the variation associated with the parameter 𝑏 (see Fig. 7). Potential causes of this performance discrepancy include regional imbalance within the training dataset, insufficient preprocessing of input variables (e.g., normalization or scaling), limitations in the network architecture (number of layers or neurons), or the loss function becoming trapped in local minima during optimization. Consequently, while the 𝑄,, method remains a reliable and preferable approach in practical applications, enhancing the general applicability of the ANN model may require increasing data diversity, incorporating regularization techniques, performing hyper parameter optimization, and/or exploring more suitable architectural alternatives (such as deeper networks, different activation functions, or ensemble-based strategies). The visualizations presented in Fig. 7 corroborate these observations. M12-9 2nd Kocaeli ScienceCongress, November 19-21, 2025 Fig8RegressionGraphs The correlation coefficients obtained for the datasets are as follows: 𝑇𝑟𝑎𝑖𝑛 𝑅=0.99997, Validation 𝑅=0.9999𝑅 ,𝑇𝑒𝑠 𝑅=1.000, and 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅=0.999𝑅, indicating that the model captures the target values with an extremely high statistical accuracy. These values suggest an almost perfect linear relationship between the model outputs and the true values, implying that the data has been learned to a very high degree of fidelity. The close agreement of 𝑅 values across the training, validation, and test sets may give the impression that the model is not overfitting; however, structural deviations observed in the surface and contour plots reveal that, despite the high correlation coefficients, the network does not fully capture the true two-variable geometry of the function. This highlights that the correlation coefficient alone is insufficient for evaluating the overall quality of a model, particularly in the learning of multivariable mathematical functions. Additional performance metrics, such as surface shape fidelity, sensitivity to individual variables, local behavior, and generalization capability, are essential. Therefore, although the ANN model produces numerically very high 𝑅 values, its inability to adequately represent variations with respect to the 𝑏 variable implies that the model cannot be considered fully accurate from a physical or mathematical standpoint. 3. Results and Discussion To assess the effectiveness of the suggested techniques for approximating the function𝑓(𝑥)=𝑥𝑙𝑜𝑔𝑥, three integral surfaces were compared. The reference was the first surface, the Exact Integral Surface, which was derived analytically. An artificial neural network (ANN) trained using the LevenbergMarquardt algorithm produced the third surface, whilst approach [9] offered a numerical approximation. According to the comparative analysis, the technique closely mimics the precise surface's overall shape and contour patterns. Although the ANN model can approximate the genuine surface in some areas, it has significant short comings when it comes to capturing the variation along the b parameter. According to Figs. 2 and 3, the ANN surface has a nearlyone-dimensional structure that varies mostly along parameter 𝑎 and shows little sensitivity to changes in 𝑏. This shows that the target function's underlying two-dimensional dependence was not fully learned by the network, most likely as a result of restrictions in the network architecture or training dataset.