Deep Learning for Solving Integral Equations: A Problem-Oriented Review with an Axiomatic Perspective

This review surveys recent deep learning approaches for solving integral equations, categorizing them into three methodological families: physics-informed embedding, spectral/topological acceleration, and hybrid symbolic–numeric frameworks. The main findings are threefold. First, these methods achieve promising empirical accuracy in oscillatory, high-dimensional, and singular-kernel settings, yet their theoretical foundations remain largely incomplete. Second, from an axiomatic perspective, most approaches lack rigorous guarantees of convergence, stability, and spectral consistency; we formulate five testable propositions that a complete theory should satisfy. Third, we identify five specific unresolved theoretical questions and outline a focused research agenda toward a mathematically rigorous theory of neural operator approximation for integral equations. The novelty of this review lies in its dual computational–axiomatic evaluation and its provision of a structured, problem-oriented framework for future investigations.