Beijing Institute of Mathematical Sciences and Applications

In this talk, we highlight some cutting-edge topological theories derived from GLMY theory in the developing new area of higher graph theory as mathematical foundation for the graph networks with dominant or hidden high-order interaction structures.

伴随着科技的飞速发展，应用和计算拓扑学的发展也日新月异，拓扑学的思想和方法正在广泛和深入地渗透到诸多应用和交叉学科领域，包括数据科学、人工智能、机器人学、分子生物学、医学科学和材料科学等，并且在很多时候成为解决问题的关键工具。本报告将概要介绍拓扑学的一些典型和最新的应用案例，解释拓扑学的理论和方法的是如何在这些应用中发挥关键作用的，以说明看起来高度抽象和深刻的拓扑学理论不仅在数学领域非常重要，在应用领域中也发挥着举足轻重的作用。

当前的多组学推断分析通常需要在数百种细胞与组织样本中执行数十种分子实验，这是一项昂贵的工作，难以在所有物种和感兴趣的生物条件下重复实施。为解决这一瓶颈，我们构建了一个低成本、高精度的统一计算框架 BioSeq2Seq，用于在有限输入条件下推断多组学功能特征。BioSeq2Seq 通过整合 DNA 序列保守性、转录活性与方向性信号，并采用可参数化的多任务架构，可从单细胞系输入推断多种分子实验结果，在组蛋白修饰、功能元件识别、基因表达和转录因子结合位点预测等任务中均取得领先性能。进一步地，为提高预测分辨率，我们构建了 MambaHM 模型，引入高效的 Mamba 状态空间结构，并融合 DNA 与 ATAC-seq 信息，实现 16 bp 分辨率的组蛋白修饰预测，并在跨细胞类型与物种的泛化能力上表现突出。两个模型共同形成一个可扩展的计算基因组学体系，为大规模基因组注释提供高效替代方案，并为精细表观遗传建模与药物研发奠定方法基础。

This talk presents Analytic Natural Language Processing (Analytic NLP) as a paradigm for cognitive computation. The central claim is that natural language understanding should go beyond end-to-end generation or classification and instead prioritize the extraction of stable, verifiable, and reusable intermediate structures that support cumulative knowledge and controllable reasoning. To this end, we introduce Lingua-Graph as a unified graph-based representation layer, built on a hierarchical predicate–function–argument formulation that explicitly encodes core linguistic substructures such as entities, facts, and nested relations. With Lingua-Graph, structural-analytic tasks—including entity recognition, coreference resolution, and relation extraction—can be reformulated as substructure identification problems, mitigating inconsistency and conflict among task-specific annotation criteria and enabling cross-task generalization. We describe the representation design, dataset construction, and baseline parsing model, and we illustrate its effectiveness through open-domain experiments, showing that Lingua-Graph captures common internal structures shared across tasks. Overall, Analytic NLP offers a complementary alternative to purely discriminative or generative pipelines by placing explicit structure at the center of cognitive computation for interpretable and scalable language understanding.

Modern artificial intelligence performs impressively in data-rich settings but still struggles to learn and adapt from only a few examples—a capability central to human intelligence. My research seeks to understand and enable data-efficient generalization, unifying principles across few-shot learning, meta-learning, in-context learning in large language models (LLMs), and adaptive agent behavior. First, I revisit few-shot learning from a foundational perspective, showing why conventional supervised learning breaks down under sparse data and how prior knowledge enables reliable adaptation. I then discuss how these principles extend to real-world scenarios such as scientific discovery and cold-start recommendation, where data are scarce, costly, or dynamically evolving. Finally, I explore how LLMs perform in-context learning and how their adaptive behaviors connect to meta-learning mechanisms. Building on these insights, I develop data-efficient, preference-adaptive agents that quickly align to user needs with minimal interaction.This talk presents a cohesive view of data-efficient intelligence and outlines future directions toward more reliable, human-like learning systems.

Networks are fundamental to understanding complex systems, characterized by many underlying entities and their intricate interactions. We contextualize evolutionary game theory and ecology niche theory into a unified framework to explain how the dynamic change of an entity is determined by its own strategy and the strategies of its interacting counterparts. We derive a system of nonlinear mixed ordinary differential equations (nMODEs) to quantify the contributions of these two types of strategies and encode them into informative, dynamic, omnidirectional, and personalized networks (idopNetworks). We implement multi-task learning (MTL) into the matrix representation of linearized nMODEs to choose a subset of the most significant entities (acting as predictors) jointly for all entities each viewed as a response. By integrating both group and elementwise sparsity, the model imposes double sparsity constraint—on regulatory edges and nonlinear features—yielding consistent edge selection and a compact, interpretable dynamical representation. In going beyond existing networking practice, idopNetworks can capture all-around interacting links, nonlinearities, and emergent properties of a complex system, which, to a larger extent, approximate the intricate and multifaceted nature of complex systems. We apply our model to learn gene regulatory idopNetworks from transcriptional data, identifying previously-unknown regulatory roles of several genes in mediating malaria infection. We perform computer simulation to validate the statistical relevance of the model. Our model provides a new insight of machine learning to analyze, model, and interpret complex data in a non-Euclidean space.

This study proposes a two-phase machine learning approach to extract interaction kernels from trajectory data of many-particle systems. The research is motivated by understanding interaction laws governing collective behaviors such as flocking birds, schooling fish, and opinion dynamics. Based on mean-field theory, as particle number N approaches infinity, the system can be described by the McKean-Vlasov equation. The method consists of two phases: Phase I employs importance sampling with adaptive sparsification for initial estimation; Phase II refines parameter estimation using the complete dataset. Results are evaluated through free energy comparison, kernel function errors, and Wasserstein distance metrics. Numerical experiments cover diverse scenarios including cubic potential, power-law repulsion-attraction, double-well potential, opinion dynamics (polarization and consensus models), and 2D radial evolution patterns. The method successfully reconstructs interaction kernels across all cases. The approach combines kernel density estimation with symbolic regression techniques, enabling identification of both smooth and discontinuous interaction functions.

相变广泛存在于无生命系统中，并且已在自然科学中得到广泛研究和应用。但是对于在生物系统中发生在微纳米级的丰富相变现象的探索相对较少。与传统相变特征类似，系统宏观属性会随着外部条件的连续变化发生不连续变化，这种变化在生物体各个尺度如蛋白质分子、细胞和组织等动态行为中起重要作用。本讲座介绍细菌发生在多尺度和多物理场的相变现象：<br>1. 细胞分化 - 细菌在形成细菌生物膜的过程中会根据环境变化分化成具有不同功能的基因表现型细胞；<br>2. 表面形貌多级演化 - 细菌生物膜在生长过程中会呈现出丰富的表面形貌结构，从而使其在极其恶劣的环境下生存；<br>3.不同基因表现型细胞群的协同运动 - 细菌在形成细菌生物膜过程中分化成的不同功能的基因表现型细胞之间相互作用，协同运动。<br>细菌的相变研究能够为慢性疾病防治和清洁能源开发提供支撑。

流体力学、传热传质学在工程与科学领域具有重要应用价值。为深入揭示流动、热传导及反常扩散等现象的内在机理，数值模拟成为一种有效的研究手段，能够显著降低实验成本。基于经典牛顿流体模型、Fourier传热定律及Fick扩散定律，本课题组发展了非线性耦合模型理论，并采用数值方法，针对强非线性耦合控制方程组开发了数值模拟方法，实现了对流场、温度场及浓度场输运过程的高效仿真，并验证了模拟方法的有效性与准确性。通过对粘弹性液滴蒸发、生物组织传热以及梳状反常扩散等具体问题的分析，深入揭示了相关动力学机制，从而为相应工业应用提供了理论依据与技术指导。

In this talk, I will present the asymptotic stability results of 2D Couette flow governed by the Navier-Stokes Boussinesq system. And we mainly give a sharp stability threshold when the Richardson number is larger than 1/4.

在增材制造领域，为提升金属材料的力学性能，常在基体中引入碳化钨作为增强颗粒，形成颗粒增强复合材料。然而，增材制造过程中增强颗粒的非均匀运动易导致团聚等问题，进而损害复合材料的力学性能与耐腐蚀性。此外，该过程中粉末熔化与冷却时间极短，传统实验手段难以对其实现有效观测与分析。近年来，数值模拟技术因其良好的灵活性、可重复性和低成本等优势，已成为研究增材制造过程中增强颗粒扩散行为的重要工具。为此，本项工作面向增材制造工艺模拟的重大需求，拟采用光滑粒子动力学（SPH）方法，发展针对激光粉末床熔融多物理场的高效高精度数值模拟算法，系统研究增强颗粒在增材制造中的扩散行为，为实现颗粒均匀分布提供工艺参数指导，并为制备高强硬、高耐蚀的颗粒增强增材制造复合材料提供理论依据与技术支撑。

气体爆轰是一类重要的流体力学问题，在国防和航空航天领域具有广泛应用。反应欧拉方程是描述爆轰问题的经典模型，属于典型的一阶双曲型偏微分方程，本报告将从偏微分方程适定性角度讨论爆轰问题的状态方程，提出一类新的非理想气体状态方程，证明在该状态方程下模型的适定性，并通过数值算例验证其有效性。进一步，基于适定性分析，构造了反应欧拉方程的熵稳定格式，该格式不仅对数学熵和热力学熵都满足熵稳定，并且保密度、压力和质量分数为正，数值算例表明格式对间断问题也有很好效果。

Space-filling designs play a central role in computer experiments, surrogate modeling, optimization, and uncertainty quantification, especially when high-fidelity simulations are expensive. Classical algebraic constructions are often restricted to special run sizes and dimensions, whereas purely algorithmic search methods can be computationally prohibitive for large-scale problems. In this work, we develop a unified framework for constructing space-filling designs based on good lattice point (GLP) sets. We first review the number-theoretic structure of GLP sets and their natural connection to Latin hypercube designs (LHDs). Building on this structure, our work enlarges the effective design search space and proposes generalized GLP-based LHDs together with efficient optimization algorithms, yielding near-maximin distance LHDs for arbitrary run sizes and dimensions. Numerical studies demonstrate that our GLP-based designs achieve better distance properties and higher computational efficiency than several commonly used design construction methods. We then extend the GLP framework to sequential, nested, and sliced designs, which are particularly suited for batched experiments, multi-fidelity simulators, and problems involving both quantitative and qualitative factors. Overall, our work provides a flexible and scalable toolbox for modern computer experiments and digital twin applications.