local spectral coarsening

本文针对复杂几何和非结构化网格上的参数化PDEs求解问题，提出了一种基于超图自适应小波算子的方法HALO，通过在超图的频域学习实现了高精度和稳定性。

This work proposes a simplex-based multidimensional persistence Laplacian framework to address the sensitivity of dimensionality reduction methods like Principal Component Analysis (PCA) to the choice of target dimension and their resulting performance instability. The approach introduces, for the first time, the multidimensional persistence Laplacian into image analysis by integrating multiscale topological spectra across multiple reduced dimensions and employing statistical aggregation to produce robust feature representations. Experimental results on the COIL20 and ETH80 datasets demonstrate that the proposed method significantly outperforms PCA baselines at moderate dimensions and exhibits greater stability across varying target dimensions, thereby enhancing both the discriminability and robustness of image representations.

Existing sheaf neural networks lack an effective hierarchical pooling mechanism for multiscale graph modeling. This work proposes HiSP, a framework that introduces, for the first time, a learnable sheaf-aware hierarchical pooling scheme. Built upon local spectral coarsening, HiSP projects fine-grained stalk features onto low-frequency modes of the cluster-wise sheaf Laplacian and preserves sheaf energy consistency via Galerkin operators. The method integrates cochain-level prolongation maps with a lifting-based sheaf Laplacian, enabling efficient batch processing within PyTorch Geometric. HiSP not only explicitly distinguishes and quantifies truncation and realization losses incurred during coarsening but also significantly enhances multiscale representation capacity while preserving the underlying sheaf structure.

该研究提出图谱神经算子(GSNO)解决非规则域上偏微分方程(PDEs)的学习问题，通过结合空间图谱分解与时间傅里叶变换，无需域扭曲或自回归展开。

This study investigates the limiting behavior of graph Laplacians as the internal connectivity within subgraphs tends to infinity, and establishes their effective representation on a coarse-grained graph. By employing resolvent convergence analysis in conjunction with spectral graph theory and cluster aggregation techniques, the authors prove that, for both undirected and directed graphs, the original Laplacian converges to an effective Laplacian defined on a reduced graph when intra-cluster edge weights diverge to infinity. Notably, in the directed case, the work reveals how the left and right nullspace structures associated with highly connected clusters critically shape the topology of the limiting graph. These findings extend spectral theory for directed graphs and provide a rigorous mathematical foundation for coarse-grained modeling of dynamical processes on networks.

本文提出了一种基于新型多尺度核框架函数逼近技术的多尺度算子学习方法，用于求解多尺度偏微分方程，并在文献中的难题上展示了其优越性。

Existing spectral operators require manual integration of higher-order structural information to obtain vertex-level representations, lacking a unified and effective mechanism. This work proposes "Collapsed Effective Operators," which automatically compress arbitrary higher-order topological information into a single vertex-level operator via the Schur complement of hierarchical Laplacian matrices. The method preserves positive semi-definiteness and, for the first time, achieves an effective collapse of higher-order structures onto the vertex level, accompanied by a spectral upper bound relative to the 0-th order Hodge Laplacian. By naturally encoding topology-mediated long-range interactions, the proposed operators significantly enhance performance in spectral clustering and signal smoothing tasks and can be readily incorporated as positional encodings within neural network architectures.

This study addresses the computational expense of traditional methods and the training inefficiency and instability of purely neural approaches for solving high-dimensional operator eigenvalue problems. To overcome these challenges, we propose the Stable Inverse Power Method Neural Network (SIPMNN). This method introduces a novel low-fidelity numerical spectral guidance mechanism, employing coarse-grid finite difference approximations to generate approximate eigenvalues as fixed shifts that guide and constrain the deep neural network training process. Experimental results on ten-dimensional benchmark problems demonstrate that SIPMNN achieves superior solution accuracy compared to purely neural baselines while reducing the required number of iterations by eight- to tenfold. Consequently, this work effectively resolves the bottlenecks associated with high-dimensional eigenvalue search difficulties and training instability.

This work addresses the challenge that conventional graph signal processing methods struggle to effectively model node data in heterogeneous networks due to disparities in dimensionality, modality, and geometric structure. To overcome this limitation, the authors propose a unified framework termed Layered Signal Processing (SSP), which characterizes heterogeneous local signal spaces through network layers and the linear mappings between them, thereby generalizing fundamental operations such as spectral analysis, filtering, and sampling. Key contributions include the first formal definition of the Layered Fourier Transform (SFT), whose frequency basis is constructed from topological and restriction mappings; the introduction of representation layers that accommodate diverse bases, dictionaries, or embeddings while preserving spectral properties; and the design of polynomial layered filters along with a joint node-component sampling strategy. Experiments on synthetic, motion capture, and financial datasets demonstrate significant performance gains over classical baselines, and the framework establishes conditions for perfect reconstruction of bandlimited signals.

Existing spectral neural operators struggle to effectively capture the local structure of physics-sensitive regions in partial differential equations (PDEs), such as abrupt material interfaces. To address this limitation, this work proposes the Edge-conditioned Spectral Operator (ESO), which innovatively incorporates edge-level local variation information to modulate global spectral mixing via a Pairwise-Variation Modal Mixer (PVMM). Furthermore, a Physics-Aware Reweighting (PAR) mechanism is introduced to adaptively enhance representations in critical regions. The proposed method achieves state-of-the-art performance across nine PDE benchmarks, significantly reducing solution errors in areas characterized by coefficient discontinuities and high-gradient flows.