arXiv:2510.06635cs.LGcs.CV2025-10被引 9

用物理神经网络导出结构信息,加速符号回归发现真实物理规律。

StruSR: Structure-Aware Symbolic Regression with Physics-Informed Taylor Guidance

  • 利用训练好的物理神经网络进行局部泰勒展开,提取导数信息作为符号表达式演化指导。
  • 通过掩码归因机制量化子树贡献,提升对物理残差和结构匹配的优化效率。
  • 适合需要可解释物理模型的科研场景,尤其在偏微分方程系统建模中表现更优。

符号回归旨在通过搜索数学公式空间,寻找可解释的解析表达式以捕捉系统行为,尤其适用于受物理定律支配的科学建模。然而,传统方法缺乏从时间序列观测中提取结构化物理先验的机制,难以获得反映系统全局行为的符号表达式。本文提出一种结构感知的符号回归框架 StruSR,利用训练好的物理信息神经网络(PINNs)从时间序列数据中提取局部结构化的物理先验。通过对训练好的 PINN 输出执行局部泰勒展开,获得基于导数的结构信息,以指导符号表达式的进化。为评估表达式各组件的重要性,引入基于掩码的归因机制,量化每个子树在结构对齐与物理残差降低中的贡献。这些敏感性得分引导遗传编程中的变异与交叉操作,保留具有高物理或结构意义的子结构,同时选择性修改信息量较少的部分。采用混合适应度函数,联合最小化物理残差与泰勒系数不匹配,确保与控制方程及由 PINN 编码的局部解析行为一致。在基准偏微分方程系统上的实验表明,与传统基线相比,StruSR 在收敛速度、结构保真度和表达式可解释性方面均有显著提升,为基于物理约束的符号发现提供了一种原则性范式。

原文摘要 · Abstract (English)

Symbolic regression aims to find interpretable analytical expressions by searching over mathematical formula spaces to capture underlying system behavior, particularly in scientific modeling governed by physical laws. However, traditional methods lack mechanisms for extracting structured physical priors from time series observations, making it difficult to capture symbolic expressions that reflect the system's global behavior. In this work, we propose a structure-aware symbolic regression framework, called StruSR, that leverages trained Physics-Informed Neural Networks (PINNs) to extract locally structured physical priors from time series data. By performing local Taylor expansions on the outputs of the trained PINN, we obtain derivative-based structural information to guide symbolic expression evolution. To assess the importance of expression components, we introduce a masking-based attribution mechanism that quantifies each subtree's contribution to structural alignment and physical residual reduction. These sensitivity scores steer mutation and crossover operations within genetic programming, preserving substructures with high physical or structural significance while selectively modifying less informative components. A hybrid fitness function jointly minimizes physics residuals and Taylor coefficient mismatch, ensuring consistency with both the governing equations and the local analytical behavior encoded by the PINN. Experiments on benchmark PDE systems demonstrate that StruSR improves convergence speed, structural fidelity, and expression interpretability compared to conventional baselines, offering a principled paradigm for physics-grounded symbolic discovery.

符号回归物理信息遗传算法可解释模型

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