arXiv subjects

Fan Gu

Publications and source records attributed to Fan Gu.

8 recordsLinked to original sources

arXiv2026

CertBind from Multimodal Connectivity to Certifiable Retrieval Decisions

Lightweight connectors make frozen multimodal encoders composable at the representation level. Deployment exposes a second problem at the level of task decisions. A connected route can expand cross-modal reach while changing an established native retrieval capability. We introduce CertBind, a multiscale theory of certifiable composition for frozen multimodal connector graphs. At the node scale, native anchors establish the exact task identification boundary under the stated chart model. At the edge scale, contract-aware conformal ranks provide graph-wide family-wise error control. At the path scale, an overlap-aware budget and clean calibration yield a finite-sample recovery radius under declared conditions. At the query scale, this radius yields a covered top-k candidate set that becomes a point certificate when its size equals k. CertBind therefore retains supported routes as Direct, sends only flagged routes to recovery, returns Certified for decisive recovery, and returns Abstain for unresolved queries. The evaluated C-MCR shared route reduced native CLIP R@1 from 0.524 to 0.290. The production fallback recovered 0.963 +- 0.002 of clean retrieval, while the passing branch recorded a no-harm value of 1.000. CertBind extends multimodal composability from connected representations to certifiable task decisions.

cs.LG↗

arXiv2026

The well-posedness of stochastic Korteweg--de Vries equations revisited

In this paper, we propose a new view, which leads to almost sure well-posedness in $H^{s}(\mathbb{R}), s\geq 0$, for studying stochastic KdV equations. Different from \cite{de1999white} or \cite{kdvmuti}, by introducing a solution space inspired by \cite{guo2009global}, we prove the local well-posedness result only under natural $H^s(\mathbb{R}), s\geq 0$ conditions parallel to deterministic KdV equations. Furthermore, just basing on the $L_x^2$ conservation law of KdV equations, we extend the solution to a global one. The well-posedness frame obtained in this paper not only reduces several restrictions of the noise kernel, but may also have crucial values when one deals with dynamical problems of stochastic KdV equations.

math.PR↗

arXiv2025

Contiguous Storage of Grid Data for Heterogeneous Computing

Structured Cartesian grids are a fundamental component in numerical simulations. Although these grids facilitate straightforward discretization schemes, their na\"{i}ve use in sparse domains leads to excessive memory overhead and inefficient computation. Existing frameworks address are primarily optimized for CPU execution and exhibit performance bottlenecks on GPU architectures due to limited parallelism and high memory access latency. This work presents a redesigned storage architecture optimized for GPU compatibility and efficient execution across heterogeneous platforms. By abstracting low-level GPU-specific details and adopting a unified programming model based on SYCL, the proposed data structure enables seamless integration across host and device environments. This architecture simplifies GPU programming for end-users while improving scalability and portability in sparse-grid and gird-particle coupling numerical simulations.

cs.CE↗

arXiv2025

Strichartz type estimates of the Airy equation

In this article, we show the necessary and sufficient conditions for the inequality $\|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}$, where $$\|u\|_{X^{s,b}}:=\|\hat{u}(\tau,\xi)\langle \xi\rangle^s\langle \tau-\xi^3\rangle^b \|_{L_{\tau,\xi}^2}. $$ Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg--de Vries equations and stochastic Korteweg--de Vries equations.

math.AP↗

arXiv2024

Stochastic Schr\"{o}dinger-Korteweg de Vries systems driven by multiplicative noises

In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in $H_x^1\times H_x^1$. To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear terms with $b\in\left(0,1/2\right)$. Then, to overcome regularity problems, we introduce a series of approximation equations with localized nonlinear terms, which are also cutted-off in both the physical and the frequency space. By limitations, these approximation equations will help us get a priori estimate in the Bourgain space and finish the proof of the global well-posedness of the initial system.

math.PR↗

arXiv2023

On the global well-posedness of stochastic Schr\"{o}dinger-Korteweg-de Vries system

In this paper, we study the global well-posedness of the stochastic S-KdV system in $H^1(\mathbb{R})\times H^1(\mathbb{R})$, which are driven by additive noises. It is difficult to show the global well-posedness of a related perturbation system even for smooth datum and stochastic forces. To overcome it, we introduce a new sequence of approximation equations, which is the key of this paper. We establish priori estimates, global well-posedness and convergences of these approximation equations, which help us to get a pathwise priori estimate of the initial system.

math.PR↗

arXiv2022

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

In this paper, we study concentration phenomena of zero-noise limits of invariant measures for stochastic differential equations defined on $\mathbb{R}^d$ with locally Lipschitz continuous coefficients and more than one ergodic state. Under some dissipative conditions, by using Lyapunov-like functions and large deviations methods, we estimate the invariant measures in neighborhoods of stable sets, neighborhoods of unstable sets and their complement, respectively. Our result illustrates that invariant measures concentrate on the intersection of stable sets where a cost functional $W(K_i)$ is minimized and the Birkhoff center of the corresponding deterministic systems as noise tends down to zero. Furthermore, we prove the large deviations principle of invariant measures. At the end of this paper, we provide some explicit examples and their numerical simulations.

math.PR↗

arXiv2021

Improved Sensitivity of Base Layer on the Performance of Rigid Pavement

The performance of rigid pavement is greatly affected by the properties of base/subbase as well as subgrade layer. However, the performance predicted by the AASHTOWare Pavement ME design shows low sensitivity to the properties of base and subgrade layers. To improve the sensitivity and better reflect the influence of unbound layers a new set of improved models i.e., resilient modulus (MR) and modulus of subgrade reaction (k-value) are adopted in this study. An Artificial Neural Network (ANN) model is developed to predict the modified k-value based on finite element (FE) analysis. The training and validation datasets in the ANN model consist of 27000 simulation cases with different combinations of pavement layer thickness, layer modulus and slab-base interface bond ratio. To examine the sensitivity of modified MR and k-values on pavement response, eight pavement sections data are collected from the Long-Term Pavement performance (LTPP) database and modeled by using the FE software ISLAB2000. The computational results indicate that the modified MR values have higher sensitivity to water content in base layer on critical stress and deflection response of rigid pavements compared to the results using the Pavement ME design model. It is also observed that the k-values using ANN model has the capability of predicting critical pavement response at any partially bonded conditions whereas the Pavement ME design model can only calculate at two extreme bonding conditions (i.e., fully bonding and no bonding).

cs.AI↗