Data-driven identification of partial differential equations (PDEs) relies on sparse regression over a candidate library of differential operators, where larger libraries inflate false positives under observation noise and smaller libraries risk missing true terms. We introduce Equivariant Operator Discovery (EqOD), a fully automatic method combining two library reduction mechanisms. When Galilean invariance is detected from trajectory data via a weak-form structural test, EqOD uses the symmetry-reduced library, eliminating terms that our Galilean exclusion result proves to be absent from the governing equation. Otherwise, it applies randomized LASSO stability selection guided by classical false-positive bounds. A residual-based fallback prevents degradation below the full-library baseline. On 8 PDEs at 4 noise levels, EqOD attains $F_1 = 1.000 \pm 0.000$ on Heat at $20\%$ noise, where WF-LASSO obtains $0.475 \pm 0.181$, official PySINDy 2.0 obtains $0.000$, and the WSINDy reimplementation obtains $0.789$. Under the strict criterion that the mean F1 difference exceeds the larger of the two standard deviations, EqOD wins 7 of 32 cells. WF-LASSO wins none, and the remaining 25 cells are ties. Across all 32 cells, EqOD outperforms PySINDy 2.0.0 in 23 of 32 cells, and all 5 PySINDy wins occur on reaction PDEs. External validation on WeakIdent and PINN-SR datasets gives $F_1 = 1.000$ on all 5 clean benchmarks. NLS, 2D, coupled-system, and cylinder-wake extensions are reported. The Galilean library reduction is proved under explicit autonomy and library assumptions. The stability-selection step is motivated by classical false-positive bounds, while formal guarantees for correlated PDE design matrices remain open.

翻译：摘要：数据驱动的偏微分方程（PDE）识别依赖于对微分算子候选库的稀疏回归，其中较大的库在观测噪声下会放大误报，而较小的库则可能遗漏真实项。我们提出等变换算子发现（Equivariant Operator Discovery, EqOD），这是一种结合两种库缩减机制的完全自动化方法。当通过弱形式结构测试从轨迹数据中检测到伽利略不变性时，EqOD使用对称性缩减库，消除我们伽利略排除结果证明在控制方程中不存在的项；否则，该方法应用基于经典误报界引导的随机LASSO稳定性选择。残差回退机制防止性能低于完整库基线。在4种噪声水平下的8个PDE上，EqOD在20%噪声的热传导方程上达到$F_1 = 1.000 \pm 0.000$，而WF-LASSO获得$0.475 \pm 0.181$，官方PySINDy 2.0获得$0.000$，WSINDy重实现获得$0.789$。在平均F1差异超过两个标准差中较大者的严格标准下，EqOD在32个单元格中赢得7个，WF-LASSO未赢得任何单元格，其余25个单元格为平局。在所有32个单元格中，EqOD在23个单元格中优于PySINDy 2.0.0，且全部5个PySINDy胜出均发生在反应PDE上。在WeakIdent和PINN-SR数据集上的外部验证显示，所有5个干净基准测试的$F_1 = 1.000$。还报告了NLS、二维、耦合系统和圆柱尾流扩展。在明确的自治性和库假设下证明了伽利略库缩减，稳定性选择步骤受经典误报界启发，而相关PDE设计矩阵的形式化保证仍有待研究。