green's function modeling

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.

To address the high computational cost of traditional PDE solvers (e.g., FEM/FVM) and the difficulty of analytically deriving Green’s functions, this paper proposes an interpretable deep operator learning framework grounded in the Green’s function integral representation. Our method decouples modeling into two neural subnetworks: a Trunk Net that learns the implicit Green’s kernel, and a Branch Net that encodes source/boundary inputs and their gradient responses. Crucially, we introduce an embedded Green’s function learning mechanism, where the integral operator is parameterized via physics-informed neural networks to tightly couple boundary and source terms. This approach directly learns the Green’s function and its spatial gradients without mesh discretization, enhancing both efficiency and generalizability. Extensive experiments on 3D heat conduction, reaction–diffusion, and Stokes equations demonstrate superior accuracy, extrapolation capability, and interpretability compared to PINNs, DeepONet, PI-DeepONet, and FNO.

Generalizing neural operators for linear partial differential equations (PDEs) on irregular geometries to unseen source terms and boundary conditions remains challenging due to mesh dependency and lack of physical priors. Method: We propose Neural Green’s Function (NGF), the first neural operator that embeds the physical prior of Green’s functions via spectral decomposition, enabling decoupled generalization over sources and boundary conditions without meshing. NGF processes voxelized point clouds to extract local geometric features, predicts spectral coefficients of the Green’s operator, and computes solutions efficiently via numerical integration. Contribution/Results: On the MCB mechanical part thermal analysis benchmark, NGF reduces mean error by 13.9% over the best baseline neural operator and accelerates inference up to 350× compared to traditional numerical solvers. It demonstrates superior generalization across diverse geometries and functional forms of sources and boundary conditions.

This work addresses the challenge of improving robustness and generalization of neural operators for learning solution operators of parametric linear partial differential equations (PDEs), particularly under out-of-distribution (e.g., fine-grid) settings and with explicit incorporation of interpretable Green’s functions. We propose the Neural Green Operator (NGO), the first neural operator architecture to explicitly embed Green’s integral formula. NGO avoids discretization-induced bias by taking a weighted average of the input function as its input representation. It employs a dual-subnetwork design—comprising a coefficient network and a basis network—to jointly parameterize and explicitly output an approximate Green’s function, enabling direct integration into preconditioners for classical PDE solvers. Experiments demonstrate that NGO matches the in-distribution accuracy of DeepONet and Fourier Neural Operators, while exhibiting significantly superior out-of-distribution generalization. Moreover, the learned Green’s functions substantially accelerate convergence of traditional iterative solvers.

This work addresses the challenge of learning and generalizing continuous experimental Green’s functions for one-dimensional parametric systems governed by unknown partial differential equations (PDEs), specifically to model seismic wave propagation within Earth’s interior from limited observational data. Method: We propose the first mesh-free, data-driven empirical Green’s function modeling framework, integrating rational neural networks with bivariate Chebyshev orthogonal expansions; further, we introduce a novel continuous interpolation paradigm for left and right singular functions on the quasimatrix manifold, enabling cross-parameter generalization and zero-shot prediction of Green’s functions associated with latent boundary-value problems. Contribution/Results: Experiments demonstrate substantial improvements in Green’s function learning accuracy and extrapolation capability on unknown PDE systems: interpolation error is reduced by an order of magnitude compared to conventional methods.

本文提出SDC-GON方法，通过奇异分解和一致性正则化解决格林函数学习中的奇异性和一致性问题，有效求解偏微分方程。

This work addresses the linear transport partial differential equation satisfied by eigenfunctions of the Koopman operator and proposes a tailored reproducing kernel construction together with an efficient approximation method. By unifying variational principles, Green’s functions, and the method of characteristics within a reproducing kernel Hilbert space (RKHS), the study establishes—for the first time—their equivalence under this framework and leverages it to construct Mercer kernels adapted to nonlinear dynamical systems. A multi-kernel learning mechanism combined with boundary-regularized convex optimization is introduced to enable joint adaptive learning of the kernel and eigenfunctions. Theoretical analysis demonstrates that the constructed kernel converges in the L² sense to the true eigenfunction, while numerical experiments confirm the method’s robustness and broad applicability, particularly in handling eigenfunction blow-up phenomena.

This study addresses accuracy and consistency issues in the numerical construction of the Karhunen–Loève expansion (KLE) arising from discretization, quadrature rules, and finite sample sizes. It establishes an algebraic equivalence between the spectral decomposition of the Fredholm integral equation and the singular value decomposition (SVD) of a weighted sample covariance matrix, thereby unifying model-driven and data-driven KLE frameworks. The work innovatively constructs the covariance function on a non-simply-connected three-dimensional toroidal domain using the shortest interior path distance, and implements the approach numerically with unstructured meshes and Gaussian quadrature. Experiments demonstrate that, in a one-dimensional benchmark problem, SVD-based eigenvalue estimates and empirical KL coefficients converge to the theoretical 𝒩(0,1) distribution. In two-dimensional irregular and three-dimensional toroidal domains, the study systematically quantifies the combined influence of discretization strategy, quadrature accuracy, and sample size on KLE reconstruction error.

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

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.