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HMML：层次化数学建模库
2025/09/10·StepFun can make mistakes, check important info.
这一段翻译一下 To enhance LLM agents’ mathematical modeling capabilities, we introduce the Hierarchical Mathematical Modeling Library (HMML), a three-level structured hierarchy designed for efficient, targeted method retrieval. Unlike conventional flat libraries, HMML explicitly captures method heterogeneity by categorizing them into distinct modeling domains (top layer), associated subdomains (middle layer), and specific method nodes (bottom layer). This structured design streamlines retrieval through progressively refined searches guided by high-level reasoning schemas tailored specifically to mathematical modeling tasks. Specifically, HMML adopts a tree structure comprising three abstraction layers, as illustrated in Figure 2. The top layer represents distinct mathematical modeling domains, the second layer corresponds to their respective subdomains, and the third layer includes specific method nodes. Formally, the hierarchical structure of HMML is represented as follows: at the highest level, the mathematical modeling domains are denoted as T = {T(1),T(2), · · · ,T(n)}. Each modeling domain subtree T(i) is further subdivided into multiple subdomains: T(i) = {T(i,1),T(i,2), · · · ,T(i,k)}. Within each subdomain T(i,j), specific method nodes N(i,j,l) are structured explicitly as tuples: N(i,j,l) = {modeling method, core idea, application}. Here, modeling method provides a high-level introduction to the mathematical modeling approach, core idea describes the fundamental principles underpinning the modeling method, and application indicates typical scenarios and delineates their application scope, such as resource allocation optimization and production scheduling. For example, in the domain of operations research (T(1) = Operations Research), the subdomain of programming theory (T(1,1) = Programming Theory) includes the specific method node N(1,1,1), which involves the modeling method of linear programming, with the core idea of optimization using linear objectives and constraints, and its application in production resource scheduling. The final mathematical modeling library features five domains (e.g., Operations Research, Optimization, Machine Learning, Prediction and Evaluation), with 17 subdomains (e.g., Programming Theory, Graph Theory, Clustering, Statistics, etc.), encompassing approximately 98 modeling methods (e.g., Linear Programming, Ant Colony Optimization, Expectation Maximization, Analytic Hierarchy Process, Kolmogorov-Smirnov Test).

为了增强 LLM 代理的数学建模能力，我们引入了 层次化数学建模库（Hierarchical Mathematical Modeling Library，HMML），这是一个三层结构化的层次体系，旨在实现高效、有针对性的方法检索。与传统的扁平化库不同，HMML 通过将方法分类为不同的建模领域（顶层）、相关子领域（中层）和具体方法节点（底层），明确地捕捉了方法的异质性。这种结构化设计通过逐步细化的搜索来简化检索过程，这些搜索由专门针对数学建模任务的高层推理模式指导。具体来说，如图 2 所示，HMML 采用树状结构，包含三个抽象层：

  • 顶层：表示不同的数学建模领域。
  • 中层：对应于各自的子领域。
  • 底层：包括具体的方法节点。

形式上，HMML 的层次结构表示如下：

  • 最高层：数学建模领域表示为 。

  • 每个建模领域子树 进一步细分为多个子领域：
    。

  • 在每个子领域 内，具体的方法节点 被明确地结构化为三元组：
    。

    • 建模方法（Modeling Method）：提供数学建模方法的高层介绍。
    • 核心思想（Core Idea）：描述建模方法背后的基本原理。
    • 应用（Application）：指示典型场景并界定其应用范围，例如资源分配优化和生产调度。

示例：

在 运筹学（Operations Research） 领域 中，编程理论（Programming Theory） 子领域 包含具体的方法节点 ，涉及 线性规划（Linear Programming） 建模方法。其核心思想是通过线性目标和约束进行优化，应用场景包括生产资源调度。

最终的数学建模库 包括五个领域（如运筹学、优化、机器学习、预测和评估），17 个子领域（如编程理论、图论、聚类、统计学等），涵盖约 98 种建模方法（如线性规划、蚁群优化、期望最大化、层次分析法、Kolmogorov-Smirnov 检验）。