LieDiscover：用于显式开放形式对称性发现的自适应符号库构建

中文摘要 AI 辅助

从数据中发现潜在对称性已成为科学发现中的一个关键挑战。现有的数据驱动对称性发现方法无法确定未知无穷小生成元的确切数量和数学形式。最近的显式方法使用预定义的函数库来表示生成元，并通过代数优化来识别它们，但这些方法往往难以捕捉涉及高阶多项式或超越函数的复杂对称性。为解决这一局限性，我们将对称性发现表述为函数库和系数的联合优化问题。我们提出了一种新颖框架，利用编码器-解码器架构动态生成符号表达式并扩展函数库。该生成过程通过强化学习进行优化，利用逐步奖励加速符号搜索空间的探索。实验表明，LieDiscover能够成功揭示涉及高阶多项式或超越函数的开放形式无穷小生成元，而现有方法对此难以处理。所发现的对称性还提升了下游偏微分方程求解和发现任务的性能。

英文摘要

Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.

发表机构

• East China Normal University（华东师范大学）
• Shanghai Jiao Tong University（上海交通大学）
• Beihang University（北京航空航天大学）
• Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
• Northeast Normal University（东北师范大学）

机构由 AI 辅助整理，请以论文原文为准。