Shakor, Susan Ridha and Omer, Israa Mustafa (2026) Data-Driven Models for Approximating Solutions to Friedholm's Type II Integrative Equations: A New Hybrid Framework with Comprehensive Theoretical Analysis. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 7 (1). ISSN 2660-5309

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Abstract

This research paper presents an original framework called:” Integrally Informed Neural Network (IINN)”, specially designed to solve Friedholm's integral equations of type II including linear and nonlinear states, integrative systems, and the uniquely weak nuclei. The framework is based on the fundamental architectural feature of integrating the Friedholm effect directly into the neural network architecture by an structural attention mechanism coupled with a wavelength embedding layer based on B-spline functions, rather than simply representing it within the error function as in traditional physically informed neural network models, allowing the network to implicitly learn the inverse effect (I-λK)⁻¹ and immediate circulation to any new coercive function without regrouping. Theoretical analysis is based on three complementary pillars: Banach's fixed point theory to ensure the existence and unity of the solution, the spectral theory of monolithic effects for convergence analysis, and Barron's comprehensive approximation theory for deriving rigid error limits dismantled into three independent sources that are error Rounding, squaring error, and optimization error, as well as proof of stability under noise. The study conducts a standardized comparison between IINN and seven classical reference methods including B-spline waves, Har waves, heterogeneous repetition, Taylor expansion, plus mass functions, Adomian analysis, and homotopic disorder, as well as the modern effects learning models DeepONet and the Fourier neural effect. Numerical results across four categories of problems with real-world geometric applications verify an improvement in accuracy by one to two orders compared to the best competing methods, with higher stability Under Noise is supported by statistical analysis with Student t-tests and confidence ranges.

Item Type:	Article
Uncontrolled Keywords:	Friedholm's Integrative Equations, Integrally Informed Neural Network, Banach's Constant Point Theory, Effect Learning, B-Spline Waves, Spectroscopy, Stability Under Noise
Subjects:	H Social Sciences
Depositing User:	admin eprints
Date Deposited:	04 Nov 2026 07:20
Last Modified:	04 Nov 2026 07:20
URI:	http://eprints.umsida.ac.id/id/eprint/17407

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