solve integral equations

This work addresses parameter identification (i.e., equation reconstruction) and out-of-interval trajectory prediction for stochastic Volterra integral equations corrupted by Gaussian noise. We propose the first deep neural network–based joint framework: a novel architecture explicitly models both the state variable and its integral term; Volterra-kernel–defined integral constraints are embedded into the loss function to enable simultaneous optimization of parameter estimation and extrapolatory prediction. Prediction reliability is quantified via 95% confidence intervals. Numerical experiments demonstrate that the method achieves high-accuracy parameter identification (mean error < 2.3%) and robust trajectory prediction across multiple noise levels—substantially outperforming conventional approaches—while exhibiting strong robustness and generalization capability.

Conventional numerical methods struggle with high-dimensional, strongly singular, fractional-order integral and integro-differential equations. Method: This paper proposes the Residual Integral Solver Network (RISN), a novel physics-informed neural network that integrates residual connections with high-accuracy numerical operators—including Gaussian quadrature kernels and fractional-order derivative matrices—while incorporating physical constraints, automatic differentiation, and joint loss optimization. Contribution/Results: RISN significantly enhances network depth adaptability and modeling capability for complex integral kernels. Across diverse benchmark problems—including one- and multi-dimensional, ordinary and partial, coupled, and fractional-order equations—RISN achieves 40–78% lower mean absolute error (MAE) than standard PINNs. It demonstrates superior stability and generalization, particularly for high-dimensional, strongly singular problems where classical numerical solvers fail.

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.

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.

Solving high-dimensional partial integro-differential equations (PIDEs) numerically remains challenging due to computational intractability and poor interpretability. To address this, we propose FEX-PG, a finite-expression method featuring a novel parameter grouping (PG) strategy that drastically reduces the number of trainable coefficients required for high-dimensional function approximation. FEX-PG explicitly approximates nonlocal integral terms via truncated Taylor series, balancing computational efficiency with enhanced accuracy. The method yields compact, physically meaningful explicit analytical solutions. Evaluated on multiple high-dimensional benchmark PIDEs, FEX-PG achieves relative errors on the order of single-precision machine epsilon (~1×10⁻⁷), substantially outperforming conventional finite element and finite difference methods as well as state-of-the-art deep learning approaches. Thus, FEX-PG simultaneously delivers high accuracy, strong interpretability, and computational feasibility.

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 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 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 absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This work addresses the high computational cost of solving high-dimensional partial integro-differential equations (PIDEs), which arises from their nonlocal jump terms. The authors propose a mesh-free iterative neural solver that reformulates PIDE solution as a recursive regression problem over the entire space-time domain. By leveraging a single-jump Monte Carlo sampling strategy, the method implicitly handles the nonlocal integral term, thereby avoiding explicit integration and full residual differentiation. Embedded within the physics-informed neural networks (PINNs) framework and augmented with an iterative learning strategy, the approach efficiently captures the global solution. The method is theoretically guaranteed to converge via contraction mapping for linear PIDEs and demonstrates high accuracy and strong scalability across multiple high-dimensional linear and nonlinear test cases.