arXiv subjects

Hongyu Liu

Publications and source records attributed to Hongyu Liu.

At least 91 records · Page 5Linked to original sources

arXiv2026

Sub-wavelength resonances in two-dimensional multi-layer elastic media

In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^ω$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^ω$ as $ω\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(ω^{-2})$ as the incident frequency $ω$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes.

math.AP↗

arXiv2025

Optimizing Generative Ranking Relevance via Reinforcement Learning in Xiaohongshu Search

Ranking relevance is a fundamental task in search engines, aiming to identify the items most relevant to a given user query. Traditional relevance models typically produce scalar scores or directly predict relevance labels, limiting both interpretability and the modeling of complex relevance signals. Inspired by recent advances in Chain-of-Thought (CoT) reasoning for complex tasks, we investigate whether explicit reasoning can enhance both interpretability and performance in relevance modeling. However, existing reasoning-based Generative Relevance Models (GRMs) primarily rely on supervised fine-tuning on large amounts of human-annotated or synthetic CoT data, which often leads to limited generalization. Moreover, domain-agnostic, free-form reasoning tends to be overly generic and insufficiently grounded, limiting its potential to handle the diverse and ambiguous cases prevalent in open-domain search. In this work, we formulate relevance modeling in Xiaohongshu search as a reasoning task and introduce a Reinforcement Learning (RL)-based training framework to enhance the grounded reasoning capabilities of GRMs. Specifically, we incorporate practical business-specific relevance criteria into the multi-step reasoning prompt design and propose Stepwise Advantage Masking (SAM), a lightweight process-supervision strategy which facilitates effective learning of these criteria through improved credit assignment. To enable industrial deployment, we further distill the large-scale RL-tuned model to a lightweight version suitable for real-world search systems. Extensive offline evaluations and online A/B tests demonstrate that our approach consistently delivers significant improvements across key relevance and business metrics, validating its effectiveness, robustness, and practicality for large-scale industrial search systems.

cs.IR↗

arXiv2025

Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models

This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.

math.AP↗

arXiv2025

On surface polariton resonance and its curvature concentration effects from 3D elastic nanorods

This paper investigates surface polariton resonance (SPR) in three-dimensional elastic metamaterials with nanorod geometry. The primary motivation is to surpass the physical limitations imposed by the quasi-static approximation for SPRs through anisotropic geometric design. The analysis boils down to analyzing the spectral properties of the matrix-valued elastic Neumann-Poincaré (NP) operator defined on the nanorod boundary. We develop novel analytical techniques and conduct a rigorous asymptotic analysis of elastic layer potential operators, specifically adapted for highly anisotropic structures. Within this framework, we derive precise asymptotic formulas for the scattered field in the quasi-static regime. A thorough examination of these expressions yields explicit resonance conditions that intricately link three fundamental parameters: elastic material parameters, wave frequency, and nanorod geometry. Furthermore, we characterize the intrinsic relationship between these parameters and the associated energy blow-up rate of the resonant field. This analysis explicitly establishes a sharp curvature concentration effect at the nanorod extremities, where field enhancement is locally maximized. Our work provides a rigorous theoretical foundation for harnessing elastic SPRs through anisotropic geometric engineering, with implications for sensing, wave focusing, and metamaterial applications.

math-ph↗

arXiv2025

Topological Braiding and Dynamic Probing of Phase Transitions at Temporal Interfaces in Non-Hermitian Synthetic Dimensions

Non-Hermitian systems give rise to distinct topological phenomena, yet their manifestations at temporal interfaces characterized by abrupt changes in system parameters remain largely unex plored. Upon an abrupt alteration of the Hamiltonian in a one-dimensional non-Hermitian sys tem,the ensuring temporal interface excites both reflected and refracted wave modes. By intro ducing a chiral-symmetric Hamiltonian, this study reveals the topological effects at such temporal interfaces. We find that the reflection and refraction coefficients exhibit a topological braiding struc ture. This structure is directly determined by the difference in the topological invariants across the interface, establishing a bulk-boundary correspondence for temporal interfaces in non-Hermitian systems. Furthermore, we propose a dynamical probe that leverages the geometric similarity of eigenstates at the temporal interface to detect topological phase transitions. These findings estab lish a fundamental connection between topological braiding and nonreciprocal dynamics at temporal interfaces, providing a platform to explore phase transition detection and nonreciprocal phenomena in time-varying non-Hermitian systems.

physics.optics↗

arXiv2025

Inverse problems for infinite-dimensional transport PDEs on Wasserstein space

We develop a foundational framework for inverse problems governed by evolutionary partial differential equations (PDEs) on the Wasserstein space of probability measures. While the forward problems for such transport-type PDEs have been extensively and intensively studied, their corresponding inverse problems--which aim to reconstruct unknown operators, cost functions, or interaction kernels from observed solution data--remain largely unexplored at this level of generality. The cornerstone of our theory is a systematic approach featuring high-order calculus on the Wasserstein space and a progressive variational scheme. This methodology is specifically designed to address the challenges inherent in inverse problems for infinite-dimensional, nonlinear, and nonlocal transport PDEs. We demonstrate the power and versatility of our theory through two canonical examples: inverse problems for both the Mean Field Control (MFC) Dynamic Programming Equation and the Mean Field Game (MFG) Master Equation. Our work provides, for the first time, a unified foundation for identifying cost functions and interaction kernels from value function data. This establishes a new and fertile field of mathematical research with significant implications for both theory and applications in stochastic control and mean field games.

math.OC↗

arXiv2025

Inversions of stochastic processes from ergodic measures of Nonlinear SDEs

We introduce and analyze a novel class of inverse problems for stochastic dynamics: Given the ergodic invariant measure of a stochastic process governed by a nonlinear stochastic ordinary or partial differential equation (SODE or SPDE), we investigate the unique identifiability of the underlying process--specifically, the recovery of its drift and diffusion terms. This stands in contrast to the classical problem of statistical inference from trajectory data. We establish unique identifiability results under several key scenarios, including cases with both multiplicative and additive noise, for both finite- and infinite-dimensional systems. Our analysis leverages the intrinsic structure of the governing equations and their quantitative relationship with the ergodic measure, thereby transforming the identifiability problem into a uniqueness issue for the solutions to the associated stationary Fokker-Planck equations. This approach reveals fundamental differences between drift and diffusion inversion problems and provides counterexamples where unique recovery fails. This work lays the theoretical foundation for a new research direction with significant potential for practical application.

math.PR↗

arXiv2025

A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws

This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in $2\times 2$ hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed observation times, we develop explicit reconstruction formulas that handle arbitrary combinations of shock and rarefaction waves through a unified equivalent shock concept. Our method constructs piecewise quadratic $C^1$ flux approximations with rigorous convergence guarantees: the approximation errors decrease quadratically with the discretization parameters for function values and linearly for derivatives under $C^{1,1}$ regularity, with enhanced cubic and quadratic convergence respectively under $C^3$ regularity. Applications to the isentropic Euler equations and the mathematically equivalent p-system in compressible fluid dynamics demonstrate the method's capability to identify complete equations of state from limited dynamic measurements, providing a systematic approach to a fundamental inverse problem in continuum mechanics.

math.AP↗

arXiv2025

Determining nonlinear balance laws in product-type domains by a single local passive boundary observation

This paper introduces an operator-theoretic paradigm for solving inverse problems in nonlinear balance laws, shifting the focus from identifying specific functional forms to recovering the input-output actions of the associated flux and source operators. It is established that a single local passive boundary observation suffices to uniquely determine realizations of these operators for systems posed on product-type domains. This framework, which encompasses dynamical regimes, reveals a holographic-type principle where macroscopic boundary data encodes microscopic dynamical information, with broad implications for fluid dynamics and reaction-diffusion systems.

math.AP↗

arXiv2025

Stress concentration via quasi-Minnaert resonance in bubble-elastic structures and applications

Stress concentration in bubble-elastic scattering scenarios has significant applications in engineering blasting and medical treatments. This study provides a comprehensive mathematical analysis of stress concentration in bubbly-elastic structures, induced by the quasi-Minnaert resonance. The quasi-Minnaert resonance manifests as two distinct wave patterns near the bubble's boundary: boundary localization and high-oscillation phenomena. We demonstrate how to leverage the quasi-Minnaert resonance to induce stress concentration in the elastic total wave field near the air bubble's boundary by appropriately selecting the incident elastic wave and high-contrast structure. The interaction between the air bubble and the elastic background couples two physical wave fields-acoustic and elastic waves-across the bubble's boundary. The intricate transmission conditions, combined with the scalar nature of acoustic waves and the vectorial nature of elastic waves, present significant analytical challenges. To address these, we employ layer potential theory and asymptotic analysis to rigorously establish the stress concentration and quasi-Minnaert resonance phenomena in a radially geometry bubble-elastic model. Extensive numerical experiments are conducted to demonstrate the stress concentration phenomenon alongside quasi-Minnaert resonance for various bubble geometries, including a unit disk, a corner domain, an apple-shaped domain in $\mathbb{R}^2$, and a ball in $\mathbb{R}^3$. The findings of this study enhance the understanding of stress concentration mechanisms and their applications in engineering blasting and medical therapies.

math-ph↗

arXiv2025

A New Quasi-Singularity Formation Mechanism for Second-order Hyperbolic Equations

This paper investigates a novel mechanism for quasi-singularity formation in both linear and nonlinear hyperbolic wave equations in two and three dimensions. We prove that over any finite time interval, there exist inputs such that the Hölder norm of the resulting wave field exceeds any prescribed bound. Conversely, the set of such almost-blowup points has vanishing measure when the aforementioned bound goes to infinity. This phenomenon thus defines a quasi-singular state, intermediate between classical singularity and regularity. Crucially, both the equation coefficients and the inputs can be arbitrarily smooth; the quasi-singularity arises intrinsically from the structure of the hyperbolic wave equation combined with specific input characteristics.

math.AP↗

arXiv2025

Follow-Your-Emoji-Faster: Towards Efficient, Fine-Controllable, and Expressive Freestyle Portrait Animation

We present Follow-Your-Emoji-Faster, an efficient diffusion-based framework for freestyle portrait animation driven by facial landmarks. The main challenges in this task are preserving the identity of the reference portrait, accurately transferring target expressions, and maintaining long-term temporal consistency while ensuring generation efficiency. To address identity preservation and accurate expression retargeting, we enhance Stable Diffusion with two key components: a expression-aware landmarks as explicit motion signals, which improve motion alignment, support exaggerated expressions, and reduce identity leakage; and a fine-grained facial loss that leverages both expression and facial masks to better capture subtle expressions and faithfully preserve the reference appearance. With these components, our model supports controllable and expressive animation across diverse portrait types, including real faces, cartoons, sculptures, and animals. However, diffusion-based frameworks typically struggle to efficiently generate long-term stable animation results, which remains a core challenge in this task. To address this, we propose a progressive generation strategy for stable long-term animation, and introduce a Taylor-interpolated cache, achieving a 2.6X lossless acceleration. These two strategies ensure that our method produces high-quality results efficiently, making it user-friendly and accessible. Finally, we introduce EmojiBench++, a more comprehensive benchmark comprising diverse portraits, driving videos, and landmark sequences. Extensive evaluations on EmojiBench++ demonstrate that Follow-Your-Emoji-Faster achieves superior performance in both animation quality and controllability. The code, training dataset and benchmark will be found in https://follow-your-emoji.github.io/.

cs.CV↗

arXiv2025

A Geometric Multigrid-Accelerated Compact Gas-Kinetic Scheme for Fast Convergence in High-Speed Flows on GPUs

Implicit methods and GPU parallelization are two distinct yet powerful strategies for accelerating high-order CFD algorithms. However, few studies have successfully integrated both approaches within high-speed flow solvers. The core challenge lies in preserving the robustness of implicit algorithms in the presence of strong discontinuities, while simultaneously enabling massive thread parallelism under the constraints of limited GPU memory. To address this, we propose a GPU-optimized, geometric multigrid-accelerated, high-order compact gas kinetic scheme (CGKS) that incorporates three key innovations: (1) a multi-color lower-upper symmetric Gauss-Seidel scheme that eliminates thread conflicts and preserves memory efficiency, serving as an implicit smoother on coarse grids; (2) a discontinuity-adaptive relaxation technique and a multigrid prolongation process, based on a discontinuous feedback factor, which dynamically stabilize shock regions without compromising convergence in smooth zones; and (3) a three-layer V-cycle geometric parallel multigrid strategy specifically tailored for unstructured meshes. Extensive tests on multi-dimensional subsonic to hypersonic flows demonstrate that our GPU-based high-performance solver achieves one to two orders of magnitude faster convergence compared to previous explicit solvers. More importantly, it preserves the shock-capturing robustness of the explicit CGKS and exhibits strong scalability on GPU architectures. This work presents a unified framework that synergistically leverages implicit acceleration and GPU optimization for high-speed flow simulations, effectively overcoming traditional trade-offs between parallelism, memory constraints, and numerical stability in high-order methods.

math.NA↗

arXiv2025

Inverse Random Source Problem for the Helmholtz Equation from Statistical Phaseless Data

This paper investigates the problem of reconstructing a random source from statistical phaseless data for the two-dimensional Helmholtz equation. The major challenge of this problem is non-uniqueness, which we overcome through a reference source technique. Firstly, we introduce some artificially added point sources into the inverse random source system and derive phase retrieval (PR) formulas for the expectation and variance of the radiated fields. This paper rigorously analyze the uniqueness and stability of the recovered statistics of the radiated fields. Afterwards, since the direct problem has a unique mild solution, by examining the expectation and variance of this solution and combined with the phase retrieval formulas, we derive the Fredholm integral equations to solve the inverse random source problem (IRSP). We prove the stability of the corresponding integral equations. To quantify the uncertainty of the random source, we utilize the Bayesian method to reconstruct the random source and establish the well-posedness of the posterior distribution. Finally, numerical experiments demonstrate the effectiveness of the proposed method and validate the theoretical results.

math.NA↗

arXiv2025

Quasi-Minnaert Resonances in High-contrast acoustic Structures and Applications to Invisibility Cloaking

This paper investigates a novel quasi-Minnaert resonance phenomenon in acoustic wave propagation through high-contrast medium in both two and three dimensions, occurring in the sub-wavelength regime. These media are characterized by physical properties significantly distinct from those of a homogeneous background. The quasi-Minnaert resonance is defined by two primary features: boundary localization, where the $L^2$-norms of the internal total field and the external scattered field exhibit pronounced concentration near the boundary, and surface resonance, marked by highly oscillatory behavior of the fields near the boundary. In contrast to classical Minnaert resonances, which are associated with a discrete spectral spectrum tied to physical parameters, quasi-Minnaert resonances exhibit analogous physical phenomena but with a continuous spectral spectrum. Using layer potential theory and rigorous asymptotic analysis, we demonstrate that the coupling between a high-contrast material structure, particularly with radial geometries, and a carefully designed incident wave is critical for inducing quasi-Minnaert resonances. Extensive numerical experiments, involving radial geometries (e.g., unit disks and spheres) and general-shaped geometries (e.g., hearts, Cassini ovals, and clovers in $\mathbb{R}^2$, and spheres in $\mathbb{R}^3$), validate the occurrence of these resonances. Furthermore, we numerically demonstrate that quasi-Minnaert resonances induce an invisibility cloaking effect in the high-contrast medium. These findings have significant implications for mathematical material science and the development of acoustic cloaking technologies.

physics.optics↗

arXiv2025

Geometrical characterizations of radiating and non-radiating elastic sources and mediums with applications

In this paper, we investigate two types of time-harmonic elastic wave scattering problems. The first one involves the scattered wave generated by an active elastic source with compact support. The second one concerns elastic wave scattering caused by an inhomogeneous medium, also with compact support. We derive several novel quantitative results concerning the geometrical properties of the underlying scatterer, the associated source or incident wave field, and the physical parameters. In particular, we show that a scatterer with either a small support or high-curvature boundary points must radiate at any frequency. These qualitative characterizations allow us to establish several local and global uniqueness results for determining the support of the source or medium scatterer from a single far-field measurement. Furthermore, we reveal new geometric properties of elastic transmission eigenfunctions. To derive a quantitative relationship between the intensity of a radiating or non-radiating source and the diameter of its support, we utilize the Helmholtz decomposition, the translation-invariant $L^2$-norm estimate for the Lamé operator, and global energy estimates. Another pivotal technical approach combines complex geometric optics (CGO) solutions with local regularity estimates, facilitating microlocal analysis near admissible $K$-curvature boundary points.

math.AP↗

arXiv2025

Follow-Your-Instruction: A Comprehensive MLLM Agent for World Data Synthesis

With the growing demands of AI-generated content (AIGC), the need for high-quality, diverse, and scalable data has become increasingly crucial. However, collecting large-scale real-world data remains costly and time-consuming, hindering the development of downstream applications. While some works attempt to collect task-specific data via a rendering process, most approaches still rely on manual scene construction, limiting their scalability and accuracy. To address these challenges, we propose Follow-Your-Instruction, a Multimodal Large Language Model (MLLM)-driven framework for automatically synthesizing high-quality 2D, 3D, and 4D data. Our \textbf{Follow-Your-Instruction} first collects assets and their associated descriptions through multimodal inputs using the MLLM-Collector. Then it constructs 3D layouts, and leverages Vision-Language Models (VLMs) for semantic refinement through multi-view scenes with the MLLM-Generator and MLLM-Optimizer, respectively. Finally, it uses MLLM-Planner to generate temporally coherent future frames. We evaluate the quality of the generated data through comprehensive experiments on the 2D, 3D, and 4D generative tasks. The results show that our synthetic data significantly boosts the performance of existing baseline models, demonstrating Follow-Your-Instruction's potential as a scalable and effective data engine for generative intelligence.

cs.CV↗

arXiv2025

Uniqueness and stability in determining the wave equation from a single passive boundary measurement

This article addresses the inverse problem of simultaneously recovering both the wave speed coefficient and an unknown initial condition (acting as the source) for the multidimensional wave equation from a single passive boundary measurement. Specifically, we establish uniqueness and Hölder stability estimates for determining these parameters in the wave equation on $\mathbb{R}^3$, where only a single boundary measurement of the solution--generated by the unknown source--is available. Our work connects to thermoacoustic and photoacoustic tomography (TAT/PAT) for the physically relevant case of piecewise constant sound speeds. We significantly relax the stringent conditions previously required for resolving this problem, extending results to general classes of piecewise constant sound speeds over inclusions with unknown locations. Moreover, we do not require decay properties in time of solutions to the wave equation, which enables our study to accommodate a much broader class of unknown sources. The approach combines low frequency-domain solution representations with distinctive properties of elliptic and hyperbolic equations.

math.AP↗