integral equation modeling

This work addresses the lack of theoretical characterization in existing kernel methods for machine learning regarding the residual structure and energy stability of multichannel signals in complex systems. The authors propose an analytical framework grounded in operator defect identities, introducing the novel concept of “telescopic energy residuals.” By integrating iterative products with a λₙ-relaxed Kaczmarz scheme, they establish admissibility conditions for residuals and derive prior energy bounds. For the first time, this framework incorporates operator defect theory into kernel methods and kernel principal component analysis (KPCA), rigorously proving explicit convergence of generalized algorithms, a residual energy decomposition theorem, and stability criteria under noise. The approach significantly extends infinite-dimensional Kaczmarz theory to broader applications in machine learning.

Learning differential equation models and their solution operators from sparse data faces dual efficiency bottlenecks: scarcity of solution samples, sparsity of observations per sample, and high computational cost during training. Method: We propose the first interpretable kernel learning framework with provable worst-case error bounds. Grounded in reproducing kernel Hilbert space theory, it jointly optimizes equation structure identification and solution operator learning via physics-informed regularization and adaptive kernel design. Error propagation analysis and nonconvex optimization enable simultaneous gains in data and computational efficiency. Results: Numerical experiments demonstrate substantial improvements over state-of-the-art methods: orders-of-magnitude reduction in required solution samples and observation points per sample; significantly lower training complexity; enhanced robustness; and accuracy gains of 1–2 orders of magnitude.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

To address the high computational cost and difficulty in balancing accuracy and efficiency arising from direct discretization of neural integral equations, this paper introduces spectral methods into the neural operator learning framework for the first time, parameterizing and learning integral operators in the frequency domain. The proposed paradigm ensures both theoretical rigor and computational efficiency: theoretically, it establishes rigorous guarantees on operator approximation capacity and numerical convergence under spectral approximation, grounded in integral equation theory and Fourier analysis; practically, it employs an optimization-driven strategy for solving second-kind integral equations, substantially reducing computational complexity while enhancing generalization and interpolation accuracy. Numerical experiments across diverse nonlinear integral equation tasks demonstrate the method’s effectiveness, stability, and superior efficiency–accuracy trade-off compared to state-of-the-art baselines.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

本文通过介绍数值线性代数在偏微分方程、机器学习和数据同化中的应用，展示了如何使用少量核心概念解决大规模稀疏系统问题。

This work addresses the challenge of learning non-expansive integral operators arising in high-dimensional Fredholm integral equations of the second kind by proposing Fredholm Integral Neural Operators (FREDINOs). FREDINOs constitute the first neural operator architecture that simultaneously guarantees universal approximation capability and strict contractivity, thereby ensuring that the learned solution operator satisfies the contraction mapping condition and enabling provable convergence of fixed-point iterations. By integrating Fredholm theory, neural operator design, and boundary integral equation frameworks, the method efficiently solves both linear and nonlinear integral equations and extends naturally to high-dimensional nonlinear elliptic partial differential equations. Numerical experiments on multidimensional benchmark problems demonstrate that FREDINOs achieve high accuracy, strong interpretability, and excellent generalization performance.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the high memory and computational costs arising from dense operators in iterative solutions of electromagnetic integral equations. Within the adaptive integral method (AIM) framework, the authors propose a fully numerical discrete operator filtering strategy that eliminates the need for analytical spectral truncation. By directly applying numerical filtering to the discretized operator, the approach achieves regularization and compression comparable to analytical filtering, while naturally integrating with fast algorithms and enhancing robustness and applicability. Combined with Calderón preconditioning, numerical experiments demonstrate that the method substantially reduces computational resource requirements for the electric field integral equation (EFIE) without compromising solution accuracy.

This work addresses the computational inefficiency associated with solving the dense matrix resulting from the discretization of the three-dimensional electric field integral equation (EFIE). To this end, the authors propose a spectral truncation filtering method based on the spherical Hankel transform. By constructing a spectral representation of the Green’s function, they apply an analytical spectral filter to the integral operator, marking the first application of this technique to operator compression and regularization for the three-dimensional EFIE. The proposed approach significantly improves the spectral distribution of both continuous and discrete operators in static and dynamic regimes, thereby substantially enhancing the convergence rate and computational efficiency of both iterative and direct solvers.