arXiv subjects

Shengyang Xu

Publications and source records attributed to Shengyang Xu.

8 recordsLinked to original sources

arXiv2026

A Pressure-Free Virtual Element Method for the Surface Stokes Problem

We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method extends the planar VEM Stokes complex to discrete surfaces, yielding a nonconforming complex with an exactly tangential, $H(\operatorname{div})$-conforming velocity space, commuting interpolation, and the correct discrete cohomology. To obtain the weak consistency needed for optimal convergence, we introduce an edge correction that preserves the complex without adding degrees of freedom; the resulting construction reduces to the original complex in the planar setting. The virtual formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct explicit macroelement representatives whose energy and load induce an alternative formulation on the VEM space, algebraically equivalent to a direct macroelement Galerkin method. The original virtual formulation and the macroelement method are stable and attain optimal first-order convergence in the broken $H^1$ norm and second-order convergence in the $L^2$ norm for the velocity. Numerical experiments illustrate these convergence rates, the role of the edge correction, and pressure robustness.

math.NA↗

arXiv2026

Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons

We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses $\mathrm{ReLU}^0+\mathrm{ReLU}$, and the second uses the degree-dependent activation $\mathrm{ReLU}^k$. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.

math.NA↗

arXiv2026

MENO: Memory-Efficient Neural Operator

We propose the Memory-Efficient Neural Operator (MENO) as a high-performance PDE neural solver based on the Manifold Function Encoder (MFE). MENO features three primary advantages: (1) MENO has a significantly smaller memory footprint and much faster training speed than other popular architectures, with the memory footprint being independent of the data resolution, and therefore holds the potential for scaling up to large-scale models. (2) MENO can accept PDE inputs of arbitrary form, including arbitrary geometric domains and arbitrary discretizations. In particular, it is capable of handling cross-geometry scenarios, i.e., where the input functions and the output solutions are defined on different manifolds. (3) MENO exhibits strong generalization capability, and achieves the best accuracy on most of the benchmarks we tested, compared with the results reported in the literature. The code is available on GitHub at https://github.com/jpzxshi/MENO, and all numerical examples in this paper can be run with a single command to reproduce the reported results.

cs.LG↗

arXiv2026

Doc-REFRAG: Rethinking Multimodal Document Retrieval-Augmented Generation

Real-world knowledge resides in multimodal documents, necessitating retrieval-augmented generation (RAG) for accurate question answering. However, existing multimodal RAG models are primarily designed for single-image or closed-document settings and exhibit limited accuracy in realistic multi-image scenarios. Moreover, processing numerous retrieved images incurs substantial computational overhead from irrelevant visual tokens. To address these challenges, we introduce DocLongRAG, a large-scale dataset of 343K question--answer pairs, each associated with an average of 37.4 retrieved images to reflect authentic RAG workflows. Building on this dataset, we propose Doc-REFRAG, a question-guided framework that compresses visual tokens into coarse chunks and selectively expands question-relevant ones via a lightweight RL-based selector. Experiments on six benchmarks show that Doc-REFRAG outperforms eleven strong baselines, achieving state-of-the-art accuracy with significantly lower inference latency. Our resources are available at https://github.com/Collab-Gen/Doc-REFRAG.

cs.IR↗

arXiv2026

Accelerating Merge with Motion Vector Difference via Filter Difference Analysis for VVenC

Merge with Motion Vector Difference (MMVD) is a key coding tool in Versatile Video Coding for improving motion prediction accuracy. However, its exhaustive search strategy imposes a significant computational burden on the encoder. To address this issue, we propose a novel fast MMVD algorithm for the VVenC encoder based on fractional motion vector filter difference analysis. By approximating the 8-tap interpolation filter with a 2-tap filter, we derive a criterion based on spatial gradients and prediction residuals for estimating the potential gain of MMVD candidates. We further generalize this criterion to accommodate both shifted integer reference samples and 2D separable filtering. To minimize the overhead of the proposed method, we introduce implementation optimizations, including symmetric offset inference and cross-shaped downsampled dot-product computation. Compared with existing fast MMVD algorithms in VVenC, our method reduces the average MMVD search ratio from 21.07\% to 11.05\% and decreases the efficiency-complexity metric $η$ from 11.79 to 7.10 under the fast preset.

eess.IV↗

arXiv2026

DocRetriever: A Plug-and-Play Framework for Multimodal Document Retrieval with Comprehensive Benchmark

Multimodal documents contain diverse elements, such as tables, figures, and layouts, which can complicate retrieval tasks. While current approaches typically combine dense visual embedding models with supervised rerankers to achieve high-precision retrieval, they face inherent limitations. First, the coarse-grained nature of dense embeddings tends to obfuscate explicit semantics, failing to leverage structurally salient information. Second, supervised reranking models suffer from generalization bottlenecks, as their performance heavily relies on domain-specific training data. Furthermore, existing benchmarks often lack diverse assessment dimensions and comprehensive relevance annotations, limiting reliable evaluation. To address these challenges, we propose DocRetriever, a plug-and-play framework. It enhances visual retrieval via a layout-aware sparse embedding technique, enabling effective hybrid encoding without the overhead of optical character recognition (OCR). We also introduce a generalizable reranker that leverages reasoning-augmented demonstrations and optimized sampling to improve accuracy in few-shot settings. Finally, we construct a new benchmark, MultiDocR, to enable more rigorous evaluation. Experiments across diverse benchmarks validate DocRetriever's superiority over state-of-the-art methods.

cs.CV↗

arXiv2025

Formalization of Optimality Conditions for Smooth Constrained Optimization Problems

Optimality conditions are central to analysis of optimization problems, characterizing necessary criteria for local minima. Formalizing the optimality conditions within the type-theory-based proof assistant Lean4 provides a precise, robust, and reusable framework essential for rigorous verification in optimization theory. In this paper, we introduce a formalization of the first-order optimality conditions (also known as the Karush-Kuhn-Tucker (KKT) conditions) for smooth constrained optimization problems by beginning with concepts such as the Lagrangian function and constraint qualifications. The geometric optimality conditions are then formalized, offering insights into local minima through tangent cones. We also establish the critical equivalence between the tangent cone and linearized feasible directions under appropriate constraint qualifications. Building on these key elements, the formalization concludes the KKT conditions through the proof of the Farkas lemma. Additionally, this study provides a formalization of the dual problem and the weak duality property.

math.OC↗

arXiv2024

Formalization of Complexity Analysis of the First-order Algorithms for Convex Optimization

The convergence rate of various first-order optimization algorithms is a pivotal concern within the numerical optimization community, as it directly reflects the efficiency of these algorithms across different optimization problems. Our goal is making a significant step forward in the formal mathematical representation of optimization techniques using the Lean4 theorem prover. We first formalize the gradient for smooth functions and the subgradient for convex functions on a Hilbert space, laying the groundwork for the accurate formalization of algorithmic structures. Then, we extend our contribution by proving several properties of differentiable convex functions that have not yet been formalized in Mathlib. Finally, a comprehensive formalization of these algorithms is presented. These developments are not only noteworthy on their own but also serve as essential precursors to the formalization of a broader spectrum of numerical algorithms and their applications in machine learning as well as many other areas.

math.OC↗