arXiv subjects

Yeyu Zhang

Publications and source records attributed to Yeyu Zhang.

12 recordsLinked to original sources

arXiv2026

Tumor boundary instability induced by pressure feedback

We consider a Hele--Shaw-type tumor growth model obtained from the incompressible limit of a porous-medium equation. The limiting model preserves the patch structure of the tumor density and therefore admits a single free-boundary formulation. We investigate how a pressure-feedback parameter associated with the homeostatic pressure affects boundary stability. Using asymptotic analysis, we study boundary perturbations of finite-radius tumors and planar traveling fronts and derive the corresponding stability criteria. For planar fronts, the stability thresholds are further shown to be bifurcation points from which nonsymmetric traveling waves emerge. Our results indicate that pressure feedback can induce boundary instability in parameter regimes where the model without pressure feedback remains stable.

math.AP↗

arXiv2026

LLM-OSDA: An Optimal-Stopping Dynamic Auction for Native Advertising in Multi-Turn LLM Conversations

LLM-native advertising embeds sponsored content directly into model-generated responses, shifting the unit of sale from a fixed slot to a moment within an evolving conversation. Existing LLM ad-auction mechanisms primarily operate within a single response, settling the winner but not the timing. The extension is nontrivial: with one native insertion opportunity per session, the stopping time depends on bids, coupling timing with allocation, so static truthfulness arguments no longer apply. We propose the LLM-based Optimal Stopping Dynamic Auction (LLM-OSDA), a dynamic cost-per-click auction that integrates Bellman optimal stopping, winner allocation, and envelope pricing. A bid-independent LLM layer estimates contextual click quality and seamlessly renders the winning ad, while bids enter only the committed auction mechanism. Under an exact Bellman oracle, the expected discounted-click allocation is monotone in each advertiser's bid, and the corresponding envelope payment makes truthful bidding weakly dominant in expectation. For practical deployment, a learned StopNet approximates the Bellman action values. We show that its decisions differ from the optimal policy only near the stopping boundary and bound the resulting incentive loss in terms of its approximation error. Experiments on a simulated conversational advertising corpus show that LLM-OSDA improves net revenue by 11 percent over the strongest fixed-timing baseline while maintaining comparable user retention. Code is at https://github.com/2025Fang2025/llm-osda.

cs.CL↗

arXiv2026

Inverse Transfer and Coherence in Rotating Stratified Flow with Clouds and Phase Transitions

Inverse energy transfer to large-scale coherent structures in idealized models of geophysical flows has been of interest for over four decades. Extensive knowledge exists regarding inverse transfer in rotating and stratified dry dynamics, characterized by the Rossby number and a single dry Froude number. The current study includes effects of water and phase changes, with dynamics characterized by the Rossby number and two Froude numbers for unsaturated and saturated environments. Using numerical computations with random forcing, inverse energy transfer is examined for a model with a Boussinesq dynamical core, incorporating water vapor and liquid water in the limit of asymptotically-fast cloud microphysics. Besides kinetic energy, total energy includes buoyant potential energies from each phase, and latent moist energy responsible for potential energy transfer at phase boundaries. The rotation and stratification terms are large and comparable, such that the dry version of the evolution equations is dominated by inverse transfer of pseudo potential vorticity(PV). For fixed Rossby and dry (unsaturated) Froude numbers, compared to dry dynamics, there is a reduction in energy transfer rate, associated with the larger Froude number of saturated regions. The upscale transfer to moist PV is influenced by nonlinear waves at lowest order resulting from nonlinear buoyancy near phase interfaces. These nonlinear waves lead to coherent updrafts and downdrafts roughly aligned with fuzzy, large-scale phase boundaries identified by the time average of a cloud indicator function. Statistical relationships between phase boundaries, updrafts/downdrafts and moist PV are explored in flow regions dominated by moist PV-vortices.

physics.flu-dyn↗

arXiv2026

Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow

In this paper, we study the advective unstable Cahn--Hilliard equation on $\mathbb T^2$ with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon Δ^2 u= Δ(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} where $u_0\in H_0^2(\mathbb T^2)$, $A,\varepsilon>0$, $a<0$, and $b\in\mathbb R$. The condition $a<0$ puts the model in an unstable phase-field regime: the nonlinear chemical potential may amplify, rather than restore, concentration fluctuations, as in spinodal decomposition. The shear term $Av_1(y)\partial_xu$ models imposed stirring along the shear direction; through mixing, it enhances dissipation and counteracts the growth driven by the unstable cubic term $Δ(au^3)$. Assuming that the shear profile has finitely many critical points and that linearly growing modes occur only in the shear direction, we prove that the $L^2$-energy converges exponentially to zero, provided $|a|$ and $\|\int_{\mathbb T} u_0(x,\cdot)\,dx\|_{L_y^2}$ are sufficiently small.

math.AP↗

arXiv2026

ATLAS-NN: Adaptive Transfer Learnable Symplectic-aware Neural Network for Long-Time Hamiltonian Dynamics

Modeling Hamiltonian systems over long temporal intervals remains a significant challenge due to intrinsic multiscale structures and rapid nonlinear transitions. While Hamiltonian Neural Networks (HNNs) incorporate geometric invariants to improve stability, they typically rely on a fixed, externally prescribed temporal structure. This lack of adaptability often leads to accumulated phase errors and degraded accuracy in systems with heterogeneous temporal scales. To address these limitations, we put forward the Adaptive Transfer Learnable Symplectic-aware Neural Network (ATLAS-NN). Our framework augments the HNN architecture with a learnable temporal scaling mechanism that parametrize a nonlinear mapping of time, automatically adapting to the system's intrinsic complexity. We propose a two-stage transfer learning strategy: the model is first trained on a short-time \textit{source} interval to identify the Hamiltonian structure and optimal temporal reparameterization; the learned scaling function is then frozen and transferred to an extended \textit{target} interval for fine-tuning. Numerical experiments on nonlinear oscillators and the chaotic Hénon--Heiles system demonstrate that ATLAS-NN provides a more efficient alternative to standard HNNs and traditional symplectic integrators, yielding nearly an order of magnitude reduction in long-time prediction error.

physics.comp-ph↗

arXiv2026

Physics-Aligned Canonical Equivariant Fourier Neural Operator under Symmetry-Induced Shifts

Neural operators approximate PDE solution maps, but they need not respect the symmetries of the governing equation. In out-of-distribution (OOD) regimes, a standard neural operator must often learn coordinate alignment and physical evolution within a single map, which can hurt generalization. We use known continuous symmetries of evolution equations on periodic domains to separate these two roles. We propose the Physics-Aligned Canonical Equivariant Fourier Neural Operator (PACE-FNO), which estimates the input frame with a Lie-algebra coordinate estimator, maps the field to a reference frame, applies a standard Fourier Neural Operator (FNO), and restores the prediction to the target frame. We train alignment and operator prediction jointly using bounded symmetry perturbations, with an optional low-dimensional refinement step that updates the estimated frame at inference. Equivariance is enforced by the input and output transformations, while the FNO architecture remains unchanged. Across 1-D and 2-D Burgers, shallow-water, and Navier-Stokes equations on periodic domains, PACE-FNO matches the in-distribution (ID) accuracy of standard neural operators and reduces out-of-distribution (OOD) relative error by up to 12x over FNO with symmetry augmentation (FNO+Aug) under translations and Galilean shifts, with smaller gains for coupled rotation-translation shifts. Ablations show that aligning the input and restoring the output frame account for most OOD gains; inference-time refinement provides a smaller correction.

cs.LG↗

arXiv2025

PIP$^2$ Net: Physics-informed Partition Penalty Deep Operator Network

Operator learning has become a powerful tool for accelerating the solution of parameterized partial differential equations (PDEs), enabling rapid prediction of full spatiotemporal fields for new initial conditions or forcing functions. Existing architectures such as DeepONet and the Fourier Neural Operator (FNO) show strong empirical performance but often require large training datasets, lack explicit physical structure, and may suffer from instability in their trunk-network features, where mode imbalance or collapse can hinder accurate operator approximation. Motivated by the stability and locality of classical partition-of-unity (PoU) methods, we investigate PoU-based regularization techniques for operator learning and develop a revised formulation of the existing POU--PI--DeepONet framework. The resulting \emph{P}hysics-\emph{i}nformed \emph{P}artition \emph{P}enalty Deep Operator Network (PIP$^{2}$ Net) introduces a simplified and more principled partition penalty that improved the coordinated trunk outputs that leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP$^{2}$ Net on three nonlinear PDEs: the viscous Burgers equation, the Allen--Cahn equation, and a diffusion--reaction system. The results show that it consistently outperforms DeepONet, PI-DeepONet, and POU-DeepONet in prediction accuracy and robustness.

cs.LG↗

arXiv2025

PAS-Net: Physics-informed Adaptive Scale Deep Operator Network

Nonlinear physical phenomena often show complex multiscale interactions; motivated by the principles of multiscale modeling in scientific computing, we propose PAS-Net, a physics-informed Adaptive-Scale Deep Operator Network for learning solution operators of nonlinear and singularly perturbed evolution PDEs with small parameters and localized features. Specifically, PAS-Net augments the trunk input in the physics informed Deep Operator Network (PI-DeepONet) with a prescribed (or learnable) locally rescaled coordinate transformation centered at reference points. This addition introduces a multiscale feature embedding that acts as an architecture-independent preconditioner which improves the representation of localized, stiff, and multiscale dynamics. From an optimization perspective, the adaptive-scale embedding in PAS-Net modifies the geometry of the Neural Tangent Kernel (NTK) associated with the neural network by increasing its smallest eigenvalue, which in turn improves spectral conditioning and accelerates gradient-based convergence. We further show that this adaptive-scale mechanism explicitly accelerates neural network training in approximating functions with steep transitions and strong asymptotic behavior, and we provide a rigorous proof of this function-approximation result within the finite-dimensional NTK matrix framework. We test the proposed PAS-Net on three different problems: (i) the one-dimensional viscous Burgers equation, (ii) a nonlinear diffusion-reaction system with sharp spatial gradients, and (iii) a two-dimensional eikonal equation. The numerical results show that PAS-Net consistently achieves higher accuracy and faster convergence than the standard DeepONet and PI-DeepONet models under a similar training cost.

physics.comp-ph↗

arXiv2025

Asymptotic Properties of a Forward-Backward-Forward Differential Equation and Its Discrete Version for Solving Quasimonotone Variational Inequalities

This paper investigates the asymptotic behavior of a forward-backward-forward (FBF) type differential equation and its discrete counterpart for solving quasimonotone variational inequalities (VIs). Building on recent continuous-time dynamical system frameworks for VIs, we extend these methods to accommodate quasimonotone operators. We establish weak and strong convergence under significantly relaxed conditions, without requiring strong pseudomonotonicity or sequential weak-to-weak continuity. Additionally, we prove ergodic convergence of the continuous trajectories, offering further insight into the long-term stability of the system. In the discrete setting, we propose a novel Bregman-type algorithm that incorporates a nonmonotone adaptive step-size rule based on the golden ratio technique. A key contribution of this work is demonstrating that the proposed method ensures strong convergence under the assumption of uniform continuity of the operator, thereby relaxing the standard Lipschitz continuity requirement prevalent in existing methods. Numerical experiments, including infinite-dimensional and non-Lipschitz cases, are presented to illustrate the improved convergence and broader applicability of the proposed approach.

math.OC↗

arXiv2025

Lie Symmetry Net: Preserving Conservation Laws in Modelling Financial Market Dynamics via Differential Equations

This paper employs a novel Lie symmetries-based framework to model the intrinsic symmetries within financial market. Specifically, we introduce Lie symmetry net (LSN), which characterises the Lie symmetries of the differential equations (DE) estimating financial market dynamics, such as the Black-Scholes equation. To simulate these differential equations in a symmetry-aware manner, LSN incorporates a Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that LSN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code is available at https://github.com/Jxl163/LSN_code.

math.AP↗

arXiv2024

A Stochastic Precipitating Quasi-Geostrophic Model

Efficient and effective modeling of complex systems, incorporating cloud physics and precipitation, is essential for accurate climate modeling and forecasting. However, simulating these systems is computationally demanding since microphysics has crucial contributions to the dynamics of moisture and precipitation. In this paper, appropriate stochastic models are developed for the phase-transition dynamics of water, focusing on the precipitating quasi-geostrophic (PQG) model as a prototype. By treating the moisture, phase transitions, and latent heat release as integral components of the system, the PQG model constitutes a set of partial differential equations (PDEs) that involve Heaviside nonlinearities due to phase changes of water. Despite systematically characterizing the precipitation physics, expensive iterative algorithms are needed to find a PDE inversion at each numerical integration time step. As a crucial step toward building an effective stochastic model, a computationally efficient Markov jump process is designed to randomly simulate transitions between saturated and unsaturated states that avoids using the expensive iterative solver. The transition rates, which are deterministic, are derived from the physical fields, guaranteeing physical and statistical consistency with nature. Furthermore, to maintain the consistent spatial pattern of precipitation, the stochastic model incorporates an adaptive parameterization that automatically adjusts the transitions based on spatial information. Numerical tests show the stochastic model retains critical properties of the original PQG system while significantly reducing computational demands. It accurately captures observed precipitation patterns, including the spatial distribution and temporal variability of rainfall, alongside reproducing essential dynamic features such as potential vorticity fields and zonal mean flows.

physics.flu-dyn↗

arXiv2021

Global existence of a non-local semilinear parabolic equation with advection and applications to shear flow

In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-Δu=|u|^p-\int_{\mathbb T^N} |u|^p \quad & \textrm{on} \quad \mathbb T^N, \\ \\ u \ \textrm{periodic} \quad & \textrm{on} \quad \partial \mathbb T^N \end{cases} \end{equation*} with initial data $u_0$ defined on $\mathbb T^N$. Here $v$ is an incompressible flow, and $\mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.

math.AP↗