ZHANG Lei张磊

Complex systems often possess multiple stable states and numerous unstable states. We introduced the original concept of the “solution landscape,” using saddle points of different indices and their connections to reveal the hierarchical structure of multiple-solution problems. We develop efficient algorithms and numerical analysis for high-index saddle dynamics, providing effective theoretical and algorithmic tools for constructing the global structure of complex systems and computing transition states and transition pathways.复杂系统往往存在多个稳定态与大量不稳定态。我们提出原创的“解景观”思想，通过计算不同指标的鞍点及其连接关系，揭示多解问题的层次结构；发展高阶鞍点动力学的高效算法及数值分析，为构建复杂系统的全局结构、过渡态和转移路径计算提供高效的理论与算法工具。

• Multistability多稳态
• Rare events稀有事件
• Saddle-point computation鞍点计算

We combine phase-field models, variational methods, and solution landscape computations to study defect structures, nucleation, and phase transitions in complex physical systems such as liquid crystals and quasicrystals. By computing stable states, saddle points, critical nuclei, and transition pathways, we seek to understand how structures form and transform and address major problems in complex physical and soft-matter systems, including crystal–quasicrystal transitions and excited states of Bose–Einstein condensates.结合相场模型、变分方法与解景观计算，研究液晶、准晶等复杂物理系统的缺陷结构、成核与相变。通过计算稳定态、鞍点、临界核和转变路径，理解结构的形成与转变，探索晶体—准晶转变及玻色–爱因斯坦凝聚体的激发态等复杂物理与软物质体系中的重大问题。

• Phase-field models相场模型
• Phase transitions成核与相变
• Liquid crystals & quasicrystals液晶与准晶

Combining dynamical systems, stochastic processes, solution landscapes, reaction–diffusion equations, and multicellular phase-field models, we study the design principles of biological networks, cell-fate decisions, embryonic and plant development, and spatiotemporal pattern formation. We seek to reveal how network topology, timescales, and noise jointly determine biological function, and integrate mathematical modeling, numerical simulation, and experimental validation to understand the roles of cell mechanics, signaling, and feedback in tissue morphogenesis.结合动力系统、随机过程、解景观、反应扩散方程与多细胞相场模型，研究生物网络的设计原理、细胞命运决定、胚胎与植物发育以及时空斑图形成。重点揭示网络拓扑、时间尺度与噪声如何共同决定生物功能，并通过数学建模、数值模拟与实验验证相结合，理解细胞力学、信号调控和反馈在组织形态发育中的作用。

• Cell fate细胞命运
• Network topology & function网络拓扑与功能
• Morphological development形态发育

We develop neural network methods for PDEs and saddle-point searches, connect optimal transport with dynamics modeling, and analyze cell states using single-cell data. We study how mathematical structure and physical or biological constraints can improve the efficiency, reliability, and interpretability of learning methods.研究机器学习与科学计算的融合，发展用于偏微分方程求解与鞍点搜索的神经网络方法，结合最优传输理论研究动力学建模，并利用单细胞数据刻画细胞状态。关注数学结构及物理、生物学约束如何提高学习方法的效率、可靠性与可解释性。

• Neural methods神经网络方法
• Optimal transport最优传输
• Single-cell data单细胞数据