As a key component of the global transition to clean energy, solar power generation requires highly accurate irradiance forecasting to support grid scheduling, renewable energy utilization, and real-time operational decision-making. To improve the accuracy, robustness, and deployment efficiency of hourly solar irradiance forecasting, this study proposes a new framework, AGCRIME-VMD-BiLSTM-SA. First, a multi-perspective feature selection strategy is adopted to identify the most influential variables from multidimensional meteorological data. Then, Arnold’s cat map is introduced into the RIME algorithm to enhance the diversity of the initial population, while an adaptive Gaussian–Cauchy mixed perturbation is incorporated to strengthen global search capability, forming the AGCRIME algorithm for optimizing the hyperparameters of variational mode decomposition (VMD). Since this parameter-search process involves repeated candidate evaluation and iterative optimization, it is computationally intensive and well suited to parallel acceleration or high-performance computing during model construction. Subsequently, the optimized VMD is used to decompose the original irradiance series into several more stable subsequences, which are further modeled by a bidirectional long short-term memory network with self-attention (BiLSTM-SA) to better capture irradiance fluctuations under complex meteorological disturbances. Experimental results show that the proposed AGCRIME-VMD-BiLSTM-SA model achieves a coefficient of determination (R2) of 0.9969. Compared with RIME-VMD-BiLSTM-SA, the mean absolute error is reduced by 26.29

Continuous-line-based halftoning represents a distinct category of image stylizing technology, where a single closed non-self-intersecting curve is used instead of pixels to render a continuous-tone image. This category includes many variants that differ significantly in how lines are generated using the employed technique. We propose a novel method for creating continuous line drawings (CLDs) by offsetting the medial axes of connected cells generated under the control of density function and directional field. Our algorithm first generates cells based on the centroidal Voronoi tessellation (CVT), and a spanning tree of the cell adjacency graph is constructed to control the generation of the final curve. During this step, we minimize an energy function that combines the CVT energy and a point alignment energy to make the density and the direction of the tree area better approximate the image tone. Secondly, a one-stroke curve is created by offsetting the medial axis of the region. To address situations involving many relatively simple polygons, we employ a discrete approach to approximate the medial axes of those polygons and develop a fast and robust technique to offset the medial axes to generate a single stroke curve. We present a range of experimental results demonstrating that the method is quick and capable of producing aesthetically pleasing curves.

In multivariate time series forecasting, incorporating oscillation-related representations has attracted increasing attention. However, most existing studies primarily emphasize amplitude-based modeling, while overlooking phase information. Under strong non-stationary disturbances, this omission may cause feature misalignment and unreliable fusion, leading to amplified error accumulation in long-horizon forecasting. To address this issue, we propose PARCNet (Phase-Aware Residual Correction Network), a framework that explicitly models envelope and phase to characterize local oscillatory states and improve long-horizon forecasting stability. Specifically, PARCNet adopts a Hilbert analytic-signal perspective to transform real-valued sequences into amplitude-phase descriptors aligned in time. A lightweight one-dimensional convolutional gating mechanism then produces time-channel adaptive weights, which drive a phase-aware residual correction to refine features in a structure-consistent manner. This design enhances representational expressiveness, corrects local oscillatory deviations, and improves the temporal consistency of cross-variable coupling. Moreover, PARCNet is implemented as a plug-and-play module that can be seamlessly integrated into existing forecasting backbones to strengthen multivariate modeling capability. Extensive experiments on twelve real-world datasets demonstrate that PARCNet consistently outperforms state-of-the-art methods, achieving higher accuracy and robustness across diverse prediction horizons and evaluation settings. The code and datasets are available at https://github.com/9527dandelion/PARCNet.

Recently, most existing point cloud frameworks tend to utilize max pooling aggregation functions to aggregate local point cloud features. However, when handling data containing local high-frequency noise such as local drop, addition, and jitter, this mechanism leads to high-frequency noise that spreads from local to global and causes severe performance degradation. To address this issue, we creatively extend the concepts from the physical field, namely electrostatic field and Coulomb forceinto geometric processing. To be specific, we treat the entire point cloud placed in an electrostatic field and each point as a probe charge and then equip this field with a set of source charges according to the structure of the cloud. We endow these two types of charges with different electric quantities, which could encode informative geometrical structural information. By analogously computing the Coulomb force between the probe charge and its corresponding source charge, we finally propose an explicit embedding called Point Geometric Coulomb Force (PGCF) for each point. Due to the deep use of the structural information of the point cloud and the fact that the electrostatic field of each source charge could not be affected by the variations of the probe charges, the PGCF has been proven to provide richer geometric information while being robust to local noises. Using the PGCF combined with point coordinates as inputs can significantly improve the performances of existing 3D point cloud feature extraction frameworks, including point convolution, graph convolution, and point transformer, without additional parameters or computational overhead, thus not affecting their inference speed. Experimental results show that integrating the PGCF into existing works brings more desirable results in a wide range of 3D point cloud analysis tasks, including classification, part segmentation, and semantic segmentation.

Multivariate time series forecasting has attracted broad attention in recent years. However, existing methods typically entangle denoising with structural learning, which makes it difficult to reliably track temporal evolution in noisy and dynamically changing sequences, thereby limiting predictive accuracy. To address this, we propose a Kalman Dynamic Filtering Network (KDFNet) for long-term timeseries forecasting. Specifically, we first embed the series into a high-dimensional state space and design a Kalman dynamic filtering module that adaptively smooths each channel and suppresses noise. We then generate forecasts using a lightweight cross-variable attention block together with a cross-time linear block. Extensive experiments on multiple public datasets show that KDFNet outperforms state-of-the-art baselines, validating its robustness and effectiveness.

Geometric partial differential equations (GPDEs), defined on Riemannian manifolds, play a fundamental role in modeling surface evolution processes in science and engineering. Traditional numerical methods, however, often require costly remeshing and recomputation when handling dynamically evolving surfaces or mesh refinements. Physics-informed neural networks (PINNs), which leverage automatic differentiation to compute derivatives and embed physical constraints directly into neural network training, offer a promising alternative. Yet, unlike classical PDEs defined on fixed domains, GPDEs involve time-varying computational surfaces, where differential operators defined on irregular discrete meshes cannot be directly handled through automatic differentiation. This poses unique challenges for applying conventional PINNs to irregular geometric domains. To address these issues, we propose an attention-based discrete physics-informed neural network (ADPINet) architecture for explicitly solving GPDEs defined on triangular mesh surfaces. The proposed method adopts a dual-discrete neural framework in both time and space, enabling direct learning on irregular spatiotemporal domains. An attention-based backbone is designed to extract global correlations among mesh vertices, while a physics-informed loss formulated via discrete differential geometry ensures physical consistency with governing equations without requiring labeled data. Once trained, ADPINet can efficiently predict solutions on refined meshes or newly introduced vertices by directly inputting their coordinates, without retraining or interpolation. Furthermore, ADPINet demonstrates shape-level generalization, being able to predict GPDE solutions for different initial surfaces with similar topological and geometric structures. Extensive numerical experiments, including comparisons with traditional numerical solvers and alternative neural architectures, demonstrate the superior accuracy, mesh-quality preservation, computational efficiency, and generalization capability of the proposed method.

A large number of 3D spectral descriptors have been proposed in the literature, which act as an essential component for 3D deformable shape matching and related applications. An outstanding descriptor should have desirable natures including high-level descriptive capacity, cheap storage, and robustness to a set of nuisances. It is, however, unclear which descriptors are more suitable for a particular application. This paper fills the gap by comprehensively evaluating nine state-of-the-art spectral descriptors on ten popular deformable shape datasets as well as perturbations such as mesh discretization, geometric noise, scale transformation, non-isometric setting, partiality, and topological noise. Our evaluated terms for a spectral descriptor cover four major concerns, i.e., distinctiveness, robustness, compactness, and computational efficiency. In the end, we present a summary of the overall performance and several interesting findings that can serve as guidance for the following researchers to construct a new spectral descriptor and choose an appropriate spectral feature in a particular application.

Horizontal and vertical lines hold significant aesthetic and psychological importance, providing a sense of order, stability, and security. This paper presents an image stylization method that quickly generates non-self-intersecting and regular continuous lines based on the Hilbert curve, a well-known space-filling curve consisting of only horizontal and vertical segments. We first calculate the grayscale threshold based on gray quantization for the original image and recursively subdivide the cells according to the density in each cell. To avoid generating new feature curves due to limited gray quantization, a recursive subdivision with probability is designed to smooth the density. Then, we utilize the rule of Hilbert curve to generate continuous lines connecting all the cells. Between different degrees of Hilbert curves, bridge curves composed of horizontal and vertical lines are constructed, which are also intersection-free, instead of a straight line linking them directly. There are two parameters provided for feasibly adjusting variate effects. The image stylization framework could be generalized to other space-filling curves like the Peano curve. Compared to existing methods, our approach can generate pleasing results quickly and is fully automated. Many results show our method is robust and effective.

Current deep functional map methods face a critical gap in map representation alignment. While shape correspondence can be represented as point-wise maps, functional maps, and complex functional maps in the spatial, spectral, and complex spectral domains, respectively, existing approaches typically integrate at most two representations, resulting in symmetry ambiguity or spatial inconsistency. In this paper, we propose the TriAlign (Triple Maps Alignment) framework, a novel three-branch deep functional map-based method that harmonizes map representations across spatial, spectral, and complex spectral domains. Additionally, we introduce an alignment loss function to align the point-wise map with the complex functional map. Extensive experiments on (near-)isometric and non-isometric datasets demonstrate the superior accuracy of our method and its generalization capabilities across different datasets and mesh discretizations. Furthermore, the new loss function improves the stability of network training.

Deep neural networks have achieved remarkable success in image denoising, yet their design is predominantly empirical without clear theoretical guidance. Recent studies have revealed connections between neural networks and physics-based differential equations, offering a reliable guideline for network designs. Nevertheless, most of the theories used to guide the design of the model are not specific to the task, the mismatch between the guiding theory and the specific task will undoubtedly undermine the suitability and strength of data-driven models in specific scientific applications. To address this, we propose a novel physics-guided learning framework by incorporating the structure of the physics-based differential equations specialized for image denoising into the advanced deep model. Our framework features an asymmetric multi-scale U-Net architecture, combining an attention-based encoder with a physics-guided decoder and loss function. Experimental results show that our approach not only surpasses state-of-the-art methods in both Gaussian and real noise removal tasks, but also reduces the model’s reliance on large datasets.

Transformer-based models and time series decomposition methods have shown remarkable effectiveness in time series forecasting tasks. However, existing methods often rely on fixed filters or kernel functions, lacking adaptability to instance- and variable-level features. Moreover, most methods focus on time-domain modeling, limiting their ability to exploit rich frequency-domain information. To tackle these challenges, we propose an advanced Transformer architecture entitled FAformer, which revisits the time series forecasting paradigm through a frequency-domain perspective. The core of FAformer is an adaptive energy decomposition module. It maps input sequences into the frequency domain and dynamically selects dominant frequency components by jointly considering instance- and variable-level features. This enables a fine-grained separation of high- and low-energy components. Then, FAformer adopts a dual-branch architecture. The high-energy component is processed through a channel-dependent mechanism that captures complex inter-variable correlations. In contrast, the low-energy component is handled by a channel-independent Flip Encoder proposed to enhance modeling generalization. This design effectively integrates the advantages of both channel modeling approaches, significantly improving the model’s representation capacity and generalization robustness. Extensive experiments on multiple real-world datasets demonstrate that FAformer significantly outperforms state-of-the-art methods in long-term forecasting tasks.

Geometric partial differential equations (geometric PDEs) are defined on manifolds in Riemannian space, specifically tailored for modeling the temporal evolution of surfaces in natural sciences and engineering. For varying initial surfaces (initial conditions), traditional numerical methods require re-solving the equation even for the same geometric PDE, which significantly hinders the efficiency of simulations. The efficient predictive capabilities of neural networks (NNs) makes them a powerful tool for solving differential equations. The solution of geometric PDEs governs the continuous evolution of the surface over time, making it challenging for most NNs to handle solution prediction for geometric PDEs with varying initial surfaces. We propose a novel neural operator-based framework for solving geometric PDEs. Once trained, our model can predict the solution of the same geometric PDE under arbitrary initial conditions (initial surfaces). To the best of our knowledge, this is the first attempt to solve geometric PDEs using the neural operator. Firstly, we employ a learned continuous Signed Distance Function (SDF) representation method (DeepSDF) to convert the initial mesh surface into an implicit level-set representation, thereby avoiding the difficulties associated with solving explicit geometric PDEs. Subsequently, by integrating the multi-scale module, we design a Multi-Scale Implicit U-Net enhanced Factorized Fourier Neural Operator (MS-IUFFNO) for solving implicit geometric PDEs. The innovative structure of the neural operator substantially improves the prediction accuracy and long-term stability for solving geometric PDEs with reduced computational complexity. In addition, we construct datasets to train neural operators to solve the mean curvature flow and Willmore flow, which are representative of geometric PDEs. Finally, a numerical benchmark is conducted to compare MS-IUFFNO to several classical neural operator models for solving the mean curvature flow and Willmore flow, where results show that our model exhibits superior performance in terms of prediction accuracy, extrapolation capability, and stability.

In recent years, deep functional maps (DFM) have emerged as a leading learning-based framework for non-rigid shape-matching problems, offering diverse network architectures for this domain. This richness also makes exploring better and novel design beliefs for existing powerful DFM components to promote performance meaningful and engaging. This paper delves into this problem and successfully produces the SEDFMNet, a simple yet highly efficient DFM pipeline. To achieve this, we systematically deconstruct the core modules of the general DFM framework and analyze key design choices in existing approaches to identify the most critical components through extensive experiments. By reassembling these crucial components, we culminate in developing our SEDFMNet, which features a simpler structure than conventional DFM pipelines while delivering superior performance. Our approach is rigorously validated through comprehensive experiments on diverse datasets, where the SEDFMNet consistently achieves state-of-the-art results, even in challenging scenarios such as non-isometric shape matching and shape matching with topological noise. Our work offers fresh insights into DFM research and opens new avenues for advancing this field.

Symmetry is widely prevalent in both natural phenomena and man-made objects. Detecting and enhancing the symmetry of shapes is crucial in fields like art and engineering. However, symmetry detection and shape symmetrization based on it for 2D deformable shapes have long been a challenge. In this paper, we propose a 2D intrinsic symmetry detection method based on functional maps. By leveraging constraints from feature-symmetric point pairs and functional maps framework, our approach formulates an optimization problem for detecting intrinsic symmetry in 2D deformable shapes. We employ spectral upsampling techniques, iteratively optimizing the functional map matrix and symmetric point-to-point mapping matrix in both frequency and spatial domains. Then we perform rapid symmetrization guided by the extracted backbone and shape symmetry. The backbone is detected by mapping skeletons, and projecting it onto a straight line segment drives automatic mesh deformation to symmetrize the 2D shape. We have tested our method on a variety of 2D shapes, and the results have demonstrated its effectiveness.

Traditional deep functional map frameworks are widely used for 3D shape matching; however, many methods fail to adaptively capture the relevant frequency information required for functional map estimation in complex scenarios, leading to poor performance, especially under significant deformations. To address these challenges, we propose a novel unsupervised learning-based framework, Deep Frequency Awareness Functional Maps (DFAFM), specifically designed to tackle diverse shape-matching problems. Our approach introduces the Spectral Filter Operator Preservation constraint, which ensures the preservation of critical frequency information. These constraints promote frequency awareness by learning a set of spectral filters and incorporating them as a loss function to jointly supervise the functional maps, pointwise maps, and spectral filters. The spectral filters are constructed using orthonormal Jacobi polynomials with learnable coefficients, enabling adaptive and efficient frequency representation. Furthermore, we propose a refinement strategy that leverages the learned spectral filters and constraints to enhance the accuracy of the final pointwise map. Extensive experiments conducted on multiple benchmark datasets demonstrate that our method outperforms state-of-the-art approaches, particularly in challenging scenarios involving non-isometric deformations and inconsistent topology.

Subseasonal to seasonal (S2S) forecasting is crucial for early weather disaster prevention. Numerical weather prediction outputs often have systematic biases that need correction. To address this issue, the deep‐learning model BiConvLSTM harnesses spatiotemporal data to correct the 2‐m air temperature forecasts from the European Centre for Medium‐Range Weather Forecasts (ECMWF) over mainland China, employing ECMWF Reanalysis v5 data as the labeled dataset. BiConvLSTM improves upon the ConvLSTM by incorporating a bidirectional arrangement. Additionally, the terrain feature is used to further boost correction accuracy. Experimental results show that our model outperforms three benchmark models, the ensemble mean (EMN) and the model output statistic based on ensemble members, EDConvLSTM across the mean absolute error (MAE), mean squared skill score (MSSS), and two anomaly correlation coefficients, the pattern correlation coefficient and temporal correlation coefficient. The differences in MAE and MSSS between BiConvLSTM and EMN gradually decrease with increasing forecast lead times but begin to widen after 16 days, indicating that BiConvLSTM enhances the predictive accuracy of S2S forecasts. To better understand the sources of model prediction errors, an error decomposition based on mean‐squared error is also conducted. Results indicate that all post‐processing models are capable of reducing the bias of the ECMWF. Deep‐learning models can lower sequence errors, whereas the BiConvLSTM effectively reduces distribution errors, particularly in high‐latitude regions. BiConvLSTM also reduces errors in the first 20 days of lead times observed in ConvLSTM by traversing sequences bidirectionally and shows enhanced performance beyond 16 days with terrain improvements compared with EMN.

In deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation-preserving term. Although commonly used regularizers include the Laplacian-commutativity term and the resolvent Laplacian commutativity term, these are limited to single-scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well-justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near-)isometric and non-isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.

The functional maps framework has achieved remarkable success in non-rigid shape matching. However, the traditional functional map representations do not explicitly encode surface orientation, which can easily lead to orientation-reversing correspondence. The complex functional map addresses this issue by linking oriented tangent bundles to favor orientation-preserving correspondence. Nevertheless, the absence of effective restrictions on the complex functional maps hinders them from obtaining high-quality correspondences. To this end, we introduce novel and powerful constraints to determine complex functional maps by incorporating multiple complex spectral filter operator preservation constraints with a rigorous theoretical guarantee. Such constraints encode the surface orientation information and enforce the isometric property of the map. Based on these constraints, we propose a novel and efficient method to obtain orientation-preserving and accurate correspondences across shapes by alternatively updating the functional maps, complex functional maps, and pointwise maps. Extensive experiments demonstrate our significant improvements in correspondence quality and computing efficiency. In addition, our constraints can be easily adapted to other functional maps-based methods to enhance their performance.