arXiv:2605.09696cs.LGcs.NE2026-05被引 2

自动生成基函数,让机器自己发现非线性系统的控制方程。

Discovery of Nonlinear Dynamics with Automated Basis Function Generation

论文配图:Discovery of Nonlinear Dynamics with Automated Basis Function Generation
图 1 · 摘自论文原文
  • 用符号回归搜索候选函数形式,再通过去重筛选构建精简基函数库。
  • 在高噪声下仍能92.8%准确恢复真实方程,优于传统方法。
  • 适合需要从数据中自动建模的科研人员,尤其复杂系统动力学研究。

从观测数据中发现控制方程仍是科学建模的核心挑战,尤其当数学结构未知时。传统稀疏识别方法(如SINDy)需人工指定基函数,若关键项缺失则模型失效;纯符号回归虽灵活但易受噪声影响,常产生复杂不稳定方程。我们提出AutoSINDy,一种发现-求解混合框架:第一阶段用PySR对数据子块进行符号回归,提取候选函数形式;第二阶段通过共线性分析分解、扩展并过滤表达式,构建最小且全面的基函数库;第三阶段使用SINDy从定制库中识别稀疏控制方程。在典型非线性系统上的实验表明,AutoSINDy在高噪声下仍能以92.8%的恢复率准确复现真实方程,相比标准SINDy和独立符号回归,预测精度更高,泛化能力更强,符号复杂度显著降低。

原文摘要 · Abstract (English)

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

非线性系统符号回归动态建模AutoML

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