Lattice metamaterials enable lightweight, multifunctional structures, yet homogenization-based evaluation of their effective properties remains computationally expensive. Neural surrogates offer speed but often lack the accuracy and stability required for engineering-grade simulations. We introduce GMT, a Geometric Multigrid Transformer - a neural solver with high numerical fidelity for fast and reliable lattice homogenization. GMT achieves architectural alignment with Geometric Multigrid (GMG) by restructuring Point Transformer V3 to operate across sparse GMG hierarchies, capturing long-range dependencies and cross-level interactions essential for multigrid convergence. To enforce physical consistency, GMT incorporates physics-aware positional encoding for strict enforcement of periodicity and predicts both the finest-level solution and multi-level residual corrections. These predictions deliver a spectrally-aligned initialization, enabling end-to-end training under physics-informed and solver-aware losses and requiring only a single GMG V-cycle refinement to reach convergence. This fusion of neural prediction and numerical rigor achieves relative residual errors of 10-5 with a 160× speedup over state-of-the-art GPU-based solvers at equivalent accuracy - particularly at high resolutions (e.g., 5123), where traditional methods become most costly. We validate GMT across mechanical and thermal domains, demonstrate robust generalization to unseen geometries and non-periodic settings, and showcase scalability to high resolutions - enabling real-time design iteration, multi-scale simulations, high-throughput material discovery, and inverse design.
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We propose a system for differentiating through solutions to geometry processing problems. Our system differentiates a broad class of geometric algorithms, exploiting existing fast problem-specific schemes common to geometry processing, including local-global and ADMM solvers. It is compatible with machine learning frameworks, opening doors to new classes of inverse geometry processing applications. We marry the scatter-gather approach to mesh processing with tensor-based workflows and rely on the adjoint method applied to user-specified imperative code to generate an efficient backward pass behind the scenes. We demonstrate our approach by differentiating through mean curvature flow, spectral conformal parameterization, geodesic distance computation, and as-rigid-as-possible deformation, examining usability and performance on these applications. Our system allows practitioners to differentiate through existing geometry processing algorithms without needing to reformulate them, resulting in low implementation effort, fast runtimes, and lower memory requirements than differentiable optimization tools not tailored to geometry processing.
The simultaneous pursuit of high fidelity, large computational throughput, and a minimal memory footprint has long constituted the central challenge in fluid simulation research. Yet state-of-the-art methods struggle to reconcile all these objectives, and often entail navigating trade-offs among them. We present Kinetic Predicted-Moment Flux Reconstruction (KPM-FR), a high-order kinetic-based scheme for low-Mach-number weakly compressible flows that advances all three fronts within a single framework. KPM-FR is a flux-form fluid flow solver rooted in the principles of the gas-kinetic scheme (GKS), deriving numerical fluxes from the locally evolved Boltzmann-BGK equation to recover Navier-Stokes (NS) solutions. Departing from the GKS and its variants, it carries out kinetic evolution entirely in moment space within the high-order flux reconstruction (FR) framework through a concise predictor-corrector scheme. This translates to two fused GPU kernels per time step, streamlining computation and confining intermediate data to on-chip memory. This design confers several practical advantages. First, compared to conventional high-fidelity lattice Boltzmann methods (LBM), the moment-based formulation reduces the per-point memory footprint by more than fivefold. Second, at matched resolutions, its high-order spatial formulation exhibits markedly lower numerical dissipation, preserving fine-scale vortical structures with greater fidelity. Combining these advantages with a near-saturated throughput exceeding 8 billion solution-point updates per second on a single consumer GPU, KPM-FR delivers large-scale fluid simulation on commodity hardware. Quantitative benchmarks confirm highorder spatial convergence and spectral-like dissipation characteristics, while validation against reference data for flow past solid bodies verifies its practical accuracy. Ultimately, we demonstrate the versatility of KPM-FR across complex geometries and large-scale turbulent flows, capturing multiscale structures with 1.8 billion solution points on a single desktop workstation.
Large language models (LLMs) are increasingly deployed in domains where errors carry high social, scientific, or safety costs. Yet standard confidence estimators, such as token likelihood, semantic similarity and multi-sample consistency, remain brittle under distribution shift, domain-specialised text, and compute limits. In this work, we present Structural Confidence, a single-pass, model-agnostic framework that enhances output correctness prediction based on multi-scale structural signals derived from a model's final-layer hidden-state trajectory. By combining spectral, local-variation, and global shape descriptors, our method captures internal stability patterns that are missed by probabilities and sentence embeddings. We conduct extensive, cross-domain evaluation across four heterogeneous benchmarks—FEVER (fact verification), SciFact (scientific claims), WikiBio-hallucination (biographical consistency), and TruthfulQA (truthfulness-oriented QA). Our Structural Confidence framework demonstrates strong performance compared with established baselines in terms of AUROC and AUPR. More importantly, unlike sampling-based consistency methods which require multiple stochastic generations and an auxiliary model, our approach uses a single deterministic forward pass, offering a practical basis for efficient, robust post-hoc confidence estimation in socially impactful, resource-constrained LLM applications.
The strategic planning and reliability of modern web services, from cloud infrastructures to e-commerce platforms, increasingly hinge on accurate long-term forecasting of high-dimensional time series. A fundamental challenge within this task is modeling the intricate point-to-point dependencies that span across both time and variable dimensions. However, many existing methods face restricted direction modeling and computational inefficiency due to their reliance on localized paradigms and Transformer architectures. To address these, we propose replacing Self-Attention with autocorrelation, achieving two key innovations: 1) We propose calculating autocorrelation across both variable and time dimensions, which is a global paradigm, to model point-to-point dependencies. 2) Our proposed Spectral Product Mechanism (SPM) optimizes traditional autocorrelation into a data-driven form suitable for deep learning. Moreover, SPM reformulates autocorrelation as spectral product and reduces the complexity from O(N2) to O(NlogN), while its Hadamard product-based correlation score matrix further reduces core computation to O(N) compared to Self-Attention's O(N2) matrix multiplication. We further propose a Generalized Spectral Product Mechanism (GSPM), which extends traditional autocorrelation by mapping input into distinct feature representations, enabling modeling of complex dependencies through cross-feature correlations. SPM and GSPM surpass current state-of-the-art (SOTA) methods on 14 authoritative benchmarks, collectively securing the top rank on 22 out of 28 metrics, while ranking 1st in time, 2nd in memory, and 2nd in parameter overhead. Source code is available at: https://github.com/lxy-PhD2022/SPM.
Despite their success in Collaborative Filtering (CF), Graph Convolutional Networks (GCNs) are often viewed as Low-pass Graph Filters (LGFs) with task-specific supervision, while the influence of the underlying graph's spectral properties on LGF performance remains underexplored. Our analysis reveals that the performance of LGFs is strongly affected by algebraic connectivity. When connectivity is strong, LGFs tend to perform well; when it is weak, their effectiveness diminishes noticeably. This spectral sensitivity highlights an important limitation of existing models. To address this limitation, we propose Graph Booster, a learnable module that adaptively improves graph connectivity by reweighting edges. Unlike heuristic preprocessing, Graph Booster identifies bottleneck edges via spectral embeddings and adjusts their weights with a monotonic network guided by a lightweight graph connectivity regularizer. Integrated into LightGCN framework, our model BoostGCN achieves improvements over state-of-the-art methods, underscoring the significance of algebraic connectivity for graph-based CF.
Source localization has attracted increasing attention in recent years due to its vital role in governing the harmful propagation. However, existing localization methods do not fully consider the temporal characteristics in propagation and struggle to leverage the continuous-time information of real-world propagation scenarios. Moreover, the aggregation ability of GNN based localization models is limited by the structural noise commonly present in complicated real-world topologies. To address these challenges, a Spectral Neural Ordinary Differential Equation (SNODE) is proposed to infer the source in real-world continuous-time scenarios. First, the forward propagation is formulated as a flow based ODE system, and the source localization problem is transformed into an inverse ODE modeling task. Second, a neural process based on a graph variational autoencoder is introduced to encode global latent propagation patterns as a conditioning variable for the ODE system. Third, a spectral graph optimization is performed to suppress topological noise by filtering out high-frequency components that degrade the quality of graph aggregation in the neural process. Comprehensive experiments demonstrate that SNODE not only outperforms the optimal baseline in real-world continuous-time propagation scenarios with an average performance improvement of 43.8%, but also achieves consistently superior performance in synthetic discrete-time datasets with an improvement of 4.5%, highlighting its strong generalization ability in different propagation settings. Our code is available at https://github.com/cgao-comp/SNODE.
Graph Transformers (GTs) are increasingly applied to social network analysis, yet their deployment is often constrained by fairness concerns. This issue is particularly critical in incomplete social networks, where sensitive attributes are frequently missing due to privacy and ethical restrictions. Existing solutions commonly generate these incomplete attributes, which may introduce additional biases and further compromise user privacy. To address this challenge, FairGE (Fair Graph Encoding) is introduced as a fairness-aware framework for GTs in incomplete social networks. Instead of generating sensitive attributes, FairGE encodes fairness directly through spectral graph theory. By leveraging the principal eigenvector to represent structural information and padding incomplete sensitive attributes with zeros to maintain independence, FairGE ensures fairness without data reconstruction. Theoretical analysis demonstrates that the method suppresses the influence of non-principal spectral components, thereby enhancing fairness. Extensive experiments on seven real-world social network datasets confirm that FairGE achieves at least a 16% improvement in both statistical parity and equality of opportunity compared with state-of-the-art baselines.
Large-scale multimodal contrastive learning has recently achieved impressive success in learning rich and transferable representations, yet it remains fundamentally limited by the uniform treatment of feature dimensions and the neglect of the intrinsic spectral structure of the learned features. Empirical evidence indicates that high-dimensional embeddings tend to collapse into narrow cones, concentrating task-relevant semantics in a small subspace, while the majority of dimensions remain occupied by noise and spurious correlations. Such spectral imbalance and entanglement undermine model generalization. We propose Spectral Disentanglement and Enhancement (SDE), a novel framework that bridges the gap between the geometry of the embedded spaces and their spectral properties. Our approach leverages singular value decomposition to adaptively partition feature dimensions into strong signals that capture task-critical semantics, weak signals that reflect ancillary correlations, and noise representing irrelevant perturbations. A curriculum-based spectral enhancement strategy is then applied, selectively amplifying informative components with theoretical guarantees on training stability. Building upon the enhanced features, we further introduce a dual-domain contrastive loss that jointly optimizes alignment in both the feature and spectral spaces, effectively integrating spectral regularization into the training process and encouraging richer, more robust representations. Extensive experiments on large-scale multimodal benchmarks demonstrate that SDE consistently improves representation robustness and generalization, outperforming state-of-the-art methods. SDE integrates seamlessly with existing contrastive pipelines, offering an effective solution for multimodal representation learning.
Graph representation learning encounters great difficulties under distribution shifts. This challenge has aroused considerable interest in graph out-of-distribution (OOD) generalization and detection, which can effectively handle covariate and semantic shifts, respectively. However, real-world graph tasks often involve complex unlabeled wild data with both covariate and semantic shifts, motivating a critical question: can we design a unified framework for joint graph OOD generalization and detection? Invariant graph learning, which extracts stable relationships between features and labels, offers a promising candidate for joint OOD generalization and detection, but faces three critical challenges (1) how to model invariant subgraphs with unlabeled data, (2) how to ensure graph representations benefit both tasks, and (3) how to integrate labeled and unlabeled data under proper invariance principles. To solve these challenges, we introduce Unified Graph Out-Of-Distribution generalization and detection framework (UniGOOD) with three tailored components. Specifically, to capture subgraphs without relying on labels, we first propose the distributional invariant subgraph generator to model subgraph conditional distributions. Next, to enable generalization and reliable detection, we propose the cross-invariant-subgraph spectral contrastive learning module to learn invariant representations from subgraph distributions. Finally, for accurate subgraph discovery across labeled and unlabeled graphs, we design the triple-population invariance regularizer to enforce the invariance principle through spectral graph theory. We prove that our method theoretically ensures accurate invariant subgraphs, enabling effective OOD generalization and detection. Experiments show that UniGOOD outperforms state-of-the-art baselines for both graph OOD generalization and detection tasks.
Accurate volumetric medical image segmentation is critical for clinical diagnosis, yet adapting two-dimensional vision foundation models (VFMs) to three-dimensional medical imaging remains challenging. While parameter-efficient fine-tuning (PEFT) methods such as LoRA and adapter-based schemes provide efficient alternatives to full fine-tuning, their geometry-agnostic parameter-space adaptations are insufficient to reconcile discrepancies induced by anisotropic voxel spacing and inter-slice discontinuities. We propose GeoSFLoRA, a geometry-constrained spectral-flow low-rank adaptation framework for efficient 2D-to-3D transfer learning. GeoSFLoRA adapts frozen 2D pretrained vision backbones by operating within a fixed low-rank spectral subspace induced by pretrained linear projections. For each adapted layer, a truncated singular value decomposition is computed once and kept frozen, preserving the original pretrained bases. A lightweight Geometry-Conditioned Encoder (GCE) extracts local volumetric descriptors, which are mapped to token-conditional residuals in the singular-value space, enabling bounded and geometry-aware spectral modulation. GeoSFLoRA consistently improves Dice and HD95 on BraTS20, MSD-Prostate, and MSD-Lung, approaching full fine-tuning performance and demonstrating an effective paradigm for 2D-to-3D medical image segmentation. The code is publicly available at https://github.com/chenbn266/GeoSFLoRA.
Spectral relaxation is widely used for large-scale combinatorial optimization due to its computational efficiency. Yet its effectiveness depends critically on the choice of graph normalization, a design decision typically made heuristically. Here, we show that normalization can be treated as a continuous optimization variable rather than a fixed preprocessing choice. Our method, Differentiable Spectral Normalization (DSN), parameterizes the spectral relaxation through a diagonal metric and maximizes the resulting lower bound via projected gradient ascent. Exact gradients are obtained through the Hellmann-Feynman theorem using only the principal eigenpair, maintaining linear complexity per iteration. On benchmark instances ranging from 10^3 to 8.4 x 10^6 nodes, DSN improves solution quality by 3-15% over static spectral methods. Its performance comes within 1-3% of state-of-the-art metaheuristics, such as simulated annealing, at up to 190x lower computational cost on large-scale instances. These results suggest that learning problem-specific relaxation geometry can substantially close the gap between spectral scalability and metaheuristic solution quality.
Sleep staging is pivotal for assessing sleep quality and clinical diagnostics. While contactless radio-frequency (RF) sensing offers an unobtrusive alternative to Polysomnography (PSG), existing methods often struggle with generalization across diverse devices and environments due to the scarcity of annotated RF data. To overcome this limitation, we propose XSensorSleep, a cross-sensor domain generalization framework. Uniquely, XSensorSleep is trained exclusively on large-scale respiratory datasets from chest and abdominal belts and is directly transferable to RF-derived respiratory signals. To bridge the domain gap, we partition the training data into multiple pseudo-domains based on data sources and sensor types, forcing the model to extract domain-invariant features from heterogeneous respiratory patterns. Furthermore, we introduce a hierarchical alignment strategy: Epoch-Level Feature Alignment (ELFA) suppresses sensor-specific morphological artifacts, while Spectral Temporal Alignment (STA) captures invariant global sleep architectures. Specifically, STA leverages the rotation-invariance of eigenvalue spectra to align whole-night transition dynamics, ensuring the model prioritizes universal physiological rhythms over device-dependent signal fluctuations. Extensive evaluations across various home and clinical RF datasets demonstrate that XSensorSleep achieves superior generalization, enabling practical, automated, and label-free non-contact sleep staging.
Hyperspectral imaging (HSI) provides detailed spectral information but is often impractical due to high cost, size, and hardware complexity. While learning-based methods attempt to recover hyperspectral images from RGB sensors, they are constrained by limited spectral measurements and noise. We propose a unified optical–computational approach that converts any standard RGB camera into a snapshot hyperspectral imaging system. Our method introduces a diffractive optical adapter that replaces the conventional lens with a diffractive lens array optimized for spectral encoding. To reconstruct hyperspectral images from the resulting measurements, we design a neural network specialized for diffractively encoded RGB data, capable of compensating for optical distortions and recovering high-quality spectra. The proposed system achieves up to 10 dB improvement in PSNR over RGB-based hyperspectral reconstruction and enhances the performance of existing state-of-theart models by up to 6 dB when used with the proposed adapter. Our results demonstrate that diffractive optical encoding combined with learned reconstruction enables practical and scalable hyperspectral imaging using commodity RGB cameras.
Attributed graph clustering has achieved remarkable success by synergistically integrating topological structures and node attributes. While subspace learning has emerged as a dominant paradigm for node partitioning, most existing methods rely on implicit low-rank constraints, which often fail to capture complex nonlinear manifolds and suffer from prohibitive computational overhead on large-scale graphs. In this paper, we propose ELSS (Explicit Low-rank Structured Subspace learning), a scalable and robust framework that transcends implicit formulations. Specifically, ELSS learns an explicit and nonlinear low-rank subspace within a graph-structured embedding space, effectively uncovering latent cluster structures. To effectively mitigate the pervasive oversmoothing issue, we introduce a homophily-aware adaptive graph filter, which dynamically calibrates smoothing intensity to preserve discriminative ego-information. Furthermore, to ensure linear scalability, we develop a PageRank-guided structural sampling strategy for anchor-based approximation, which identifies pivotal landmarks based on their global topological prestige. Theoretical analysis guarantees that ELSS effectively mitigates spectral collapse while maintaining a linear complexity. Extensive experiments on diverse benchmarks demonstrate that ELSS consistently delivers superior clustering accuracy over state-of-the-art methods.
Pansharpening aims to fuse a high-resolution panchromatic (PAN) image with a low-resolution multispectral (MS) image to generate a highresolution multispectral output that preserves both fine spatial details and faithful spectral responses. However, enhancing spatial textures without introducing spectral distortion remains challenging due to the inherent spectral–spatial trade-off. To address this issue, we propose a two-stage pansharpening framework that tackles the problem from both modeling and optimization perspectives. In the first stage, we formulate pansharpening as spectral-prior conditioned residual diffusion, where a stable spectral base constrains the generation process and allows the diffusion model to focus on PAN-guided high-frequency details, leading to improved training stability and reduced spectral drift. To better capture the coupled spatial and frequency characteristics of PAN–MS fusion, we adopt a complex-valued denoising network to enhance spectral–spatial interaction modeling. In the second stage, to bridge the gap between distortionoriented training objectives and practical quality preferences, we introduce Group Relative Preference Optimization (GRPO) to fine-tune the diffusion model using multi-objective preference signals, explicitly balancing spectral fidelity, texture sharpness, and perceptual quality. Extensive experiments on standard benchmarks demonstrate that the proposed method achieves a more favorable trade-off between fidelity and perceptual quality compared to competitive end-to-end and diffusionbased approaches.
Time series forecasting aims to predict future values using historical observations, which is crucial for many practical applications with complex temporal dynamics. Recent frequency-domain forecasting methods have utilized spectral representations to model periodicity, but they usually rely on an implicit assumption: periodic components and aperiodic residuals are orthogonal in the frequency domain. However, in practice, this assumption often does not hold due to issues such as spectral leakage, superposition of multiple periods, and the gradual periodic changes. This paper re-examines frequency-domain modeling from the perspective of spectral non-identifiability and proposes a Spectral Memory Learner with Dual Retrieval (SMLDR) for time series forecasting. Instead of modeling periodicity in the frequency dimension, SMLDR represents periodic components as reusable spectral prototypes stored in learnable memory. The dual retrieval mechanism extracts historical periodic components and predicts them in the frequency domain, while a lightweight time-domain branch is used to model aperiodic dynamics to predict future residuals.Extensive experiments on multiple time series forecasting benchmarks show that SMLDR consistently achieves excellent performance, and the learnable memory can capture complex periodicity.
Over-smoothing (OSM) and over-squashing (OSQ) are two fundamental phenomena that limit the performance of Graph Neural Networks (GNNs), yet a unified spectral-spatial understanding of these phenomena remains underexplored. In this paper, we adopt polynomial spectral filters as an analytical tool to establish a unified spectral-spatial framework for graph convolution and systematically characterize the effect of the polynomial order k on information propagation in GNNs. Within this framework, we reveal an intrinsic trade-off induced by the polynomial order. Specifically, higher-order filters enhance spectral expressiveness and alleviate OSM caused by the dominant low-frequency components. However, they also expand the spatial receptive field, thereby intensifying information compression and increasing the risk of OSQ. Based on this analysis, we provide a principled guideline for selecting the polynomial order and propose a Quadratic Spectral Graph Convolution Network (QS-GCN) for graph classification. Experiments demonstrate the effectiveness and robustness of the proposed method.
Temporal embeddings play a crucial role in dynamic graph neural networks (DGNNs) by capturing the temporal dynamics of interactions. However, existing Random Fourier Feature (RFF)-based methods in DGNNs directly sample Fourier frequencies from a fixed, data-independent distribution, neglecting the temporal characteristics of dynamic graphs and thereby limiting representational capacity. We propose DeepSTE, a deep spectral temporal embedding framework for dynamic graphs. DeepSTE learns RFF representations via Monte Carlo importance sampling with a tractable proposal distribution (e.g., Gaussian) to approximate the feature map of a shift-invariant or positive-definite kernel whose latent spectral density is analytically intractable. DeepSTE adopts a data-dependent scale parameter, estimated from interaction intervals, to construct the frequency proposal distribution reflecting time–frequency uncertainty. The frequency DNN and the importance-weighting DNN, initialized from the proposal distribution, are jointly optimized to model the importance-sampled spectral representation and learn adaptive temporal features. Experiments demonstrate the effectiveness of DeepSTE on dynamic link prediction and node classification tasks, while also revealing insights such as temporal embedding decay and accelerated convergence.
Unsupervised learning of neural operators is constrained by numerical instability, causing predictions to diverge in long-horizon rollouts. To address this, we present a physics-constrained unsupervised neural operator for long-horizon PDE learning on generalized geometries (UNOP). This framework replaces differential constraints with integral consistency for stable, label-free learning. Unlike prior works, UNOP is built upon Latent Integral Physics Embedding (LIPE), which enforces physical consistency through integral constraints. To extend integral formulations to generalized geometries, the Geometry-Agnostic Latent Adapter (GALA) projects them onto a unified latent grid of PDE inputs, providing a regularized domain for spatial integral evaluation. Based on this shared embedding, the Gated Spectral Evolution Operator (GSEO) performs stable temporal integration while retaining spatial regions with sharp gradients and fine-scale structures, with the evolution constrained by the LIPE objective. Experiments on 1D, 2D, and 3D benchmarks show UNOP outperforms state of the art methods, reducing error accumulation by up to 60% in 20-step rollouts. Code is available at https://github.com/chengxinrui/UNOP.