Quan Zhao

Yuntao Wang, Yanghe Pan, Zhou Su et al.

With the rapid advancement of large models (LMs), the development of general-purpose intelligent agents powered by LMs has become a reality. It is foreseeable that in the near future, LM-driven general AI agents will serve as essential tools in production tasks, capable of autonomous communication and collaboration without human intervention. This paper investigates scenarios involving the autonomous collaboration of future LM agents. We review the current state of LM agents, the key technologies enabling LM agent collaboration, and the security and privacy challenges they face during cooperative operations. To this end, we first explore the foundational principles of LM agents, including their general architecture, key components, enabling technologies, and modern applications. We then discuss practical collaboration paradigms from data, computation, and knowledge perspectives to achieve connected intelligence among LM agents. After that, we analyze the security vulnerabilities and privacy risks associated with LM agents, particularly in multi-agent settings, examining underlying mechanisms and reviewing current and potential countermeasures. Lastly, we propose future research directions for building robust and secure LM agent ecosystems.

Harald Garcke, Robert Nürnberg, Quan Zhao

We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with boundary. The presented scheme is based on a new geometric partial differential equation (PDE) that combines an evolution equation for the mean curvature with a separate equation that prescribes the tangential velocity. The mean curvature is used to determine the normal velocity within the gradient flow structure, thus guaranteeing an unconditional energy stability for the discrete solution upon suitable discretization. We introduce a novel weak formulation for this geometric PDE, in which different types of boundary conditions can be naturally enforced. We further discretize the weak formulation to obtain a fully discrete parametric finite element method, for which well-posedness can be rigorously shown. Moreover, the constructed scheme admits an unconditional stability estimate in terms of the discrete energy. Extensive numerical experiments are reported to showcase the accuracy and robustness of the proposed method for computing Willmore flow of both curves in $\mathbb{R}^2$ and surfaces in $\mathbb{R}^3$.

Harald Garcke, Robert Nürnberg, Quan Zhao

We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous curvature effects. The axisymmetric setting allows us to formulate our schemes in terms of the generating curve of the considered surface. We propose a novel weak formulation, that combines an evolution equation for the surface's mean curvature and the curvature identity of the generating curve. The mean curvature is used to describe the gradient flow structure, which enables an unconditional stability result for the discrete solutions. The generating curve's curvature, on the other hand, describes the surface's in-plane principal curvature and plays the role of a Lagrange multiplier for an equidistribution property on the discrete level. We introduce two fully discrete schemes and prove their unconditional stability. Numerical results are provided to confirm the convergence, stability and equidistribution properties of the introduced schemes.

Harald Garcke, Robert Nürnberg, Quan Zhao

We present a unified finite element method for the dynamics of fluidic biomembranes. The model is governed by the Navier--Stokes equations in the bulk coupled to the surface Navier--Stokes equations on the evolving biomembrane surface, with bending forces arising from the Willmore energy. By allowing the bulk mesh velocity to be independent of the fluid velocity and permitting a free tangential surface velocity, we are able to derive a unified weak formulation of the coupled bulk-surface Navier--Stokes system. To address the bending force, we consider an evolution equation for the curvature and propose a surface arbitrary Lagrangian--Eulerian (ALE) weak formulation. Discretization with either fitted or unfitted finite elements leads to well-posed fully discrete linear schemes that are unconditionally energy stable. We present a variety of numerical examples to demonstrate the favourable properties of the proposed methods.

Xiaoxiao Liu, Quan Zhao

We develop a minimizing-movement framework for parametric finite element approximations of geometric gradient flows with admissible tangential motion. At each time step, the discrete variational problem combines a metric dissipation term for the normal displacement with a surface Dirichlet energy. The metric determines the normal geometric evolution: the $L^2(Γ)$ metric gives mean curvature flow, while the $H^{-1}(Γ)$ metric gives surface diffusion flow. Tangential velocity is selected independently through weak constraints on the deformation map. The central structural condition is admissibility, namely, that the identity map satisfies the constraint. This condition keeps the identity map available as a comparison function and yields the natural stability estimate. The framework recovers the classical Barrett--Garcke--Nürnberg (BGN) scheme from the unconstrained formulation and the dual minimal-deformation-rate (MDR) scheme from the MDR constraint. We further introduce two new admissible variants: an admissible BGN scheme and a relaxed MDR scheme. For the resulting fully discrete schemes, we prove existence and uniqueness under natural nondegeneracy assumptions and establish unconditional energy stability. Numerical experiments compare the admissible and classical schemes and illustrate their stability properties and mesh-quality behavior.

Xiaoxiao Liu, Quan Zhao

We propose a structure-preserving parametric finite element method for the constrained Helfrich flow of closed curves and surfaces. The proposed method is based on a two-stage velocity-splitting strategy. In the first stage, the normal velocity is computed from a curvature evolution equation, with the volume and surface area constraints imposed softly in terms of the normal velocity. This step approximates the gradient-flow structure of the Helfrich flow, and its fully discrete parametric finite element approximation leads to a linear system and yields an unconditional energy dissipation estimate at the fully discrete level. In the second stage, the surface mesh is updated by combining the computed normal velocity with a BGN-type tangential velocity. A time-weighted interface normal and an area-correction multiplier are also used to enforce exact preservation of the enclosed volume and surface area. This correction step leads to a nonlinear system, which can be efficiently solved by Newton iteration. The resulting method simultaneously achieves energy decay, exact geometric conservation, and good mesh quality. Numerical experiments for two-dimensional curves and three-dimensional surfaces, including nonsmooth initial data and nonzero spontaneous curvature, are presented to demonstrate the accuracy, robustness, and structure-preserving properties of the proposed method.