arXiv:2507.01466math-phcs.LG2025-07被引 1

从数据中自动发现多维物理场的张量方程,突破传统方法只能处理标量的局限。

Symbolic identification of tensor equations in multidimensional physical fields

  • 用宿主-质粒结构表示张量方程,结合遗传信息保留策略增强演化稳定性。
  • 在噪声数据和小样本下仍能准确恢复目标方程,且支持压缩与不可压缩流场景。
  • 创新引入维度同质性校验和张量线性回归,提升物理合理性与系数优化效率。

近年来,数据驱动方法在从仿真或实验数据中发现控制方程方面展现出巨大潜力。然而,现有方法大多局限于标量方程,极少能识别张量关系。本文提出一种通用的数据驱动框架——符号化张量方程识别(SITE),其核心思想是借鉴多维基因表达编程(M-GEP)的宿主-质粒结构来表示张量方程。为提高进化过程的鲁棒性,SITE采用遗传信息保留策略。此外,SITE在传统进化算法基础上引入两项关键创新:一是引入维度同质性检查,缩小搜索空间并剔除物理上不合理的表达式;二是以张量线性回归替代传统线性缩放,显著提升数值系数优化效率。我们在两个基准场景中验证了SITE,结果表明其能从合成数据中准确恢复目标方程,且对噪声和小样本具有强鲁棒性。进一步地,将SITE应用于分子模拟数据,直接识别本构关系,无需依赖宏观本构模型。该方法可适应压缩与不可压缩流动条件,并成功识别出相应的宏观形式,展现出数据驱动发现张量方程的巨大潜力。

原文摘要 · Abstract (English)

Recently, data-driven methods have shown great promise for discovering governing equations from simulation or experimental data. However, most existing approaches are limited to scalar equations, with few capable of identifying tensor relationships. In this work, we propose a general data-driven framework for identifying tensor equations, referred to as Symbolic Identification of Tensor Equations (SITE). The core idea of SITE--representing tensor equations using a host-plasmid structure--is inspired by the multidimensional gene expression programming (M-GEP) approach. To improve the robustness of the evolutionary process, SITE adopts a genetic information retention strategy. Moreover, SITE introduces two key innovations beyond conventional evolutionary algorithms. First, it incorporates a dimensional homogeneity check to restrict the search space and eliminate physically invalid expressions. Second, it replaces traditional linear scaling with a tensor linear regression technique, greatly enhancing the efficiency of numerical coefficient optimization. We validate SITE using two benchmark scenarios, where it accurately recovers target equations from synthetic data, showing robustness to noise and small sample sizes. Furthermore, SITE is applied to identify constitutive relations directly from molecular simulation data, which are generated without reliance on macroscopic constitutive models. It adapts to both compressible and incompressible flow conditions and successfully identifies the corresponding macroscopic forms, highlighting its potential for data-driven discovery of tensor equation.

张量方程数据驱动符号回归物理信息

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