Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm's performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.

翻译：从数据中识别偏微分方程（PDEs）对于理解自然现象的控制机制至关重要，但这一任务仍具挑战性。我们提出了ARGOS框架的扩展版本ARGOS-RAL，该方法利用稀疏回归与递归自适应lasso算法，从有限的先验知识中自动识别PDEs。我们的方法实现了偏导数的自动计算、候选函数库的构建以及稀疏模型的估计。我们严格评估了ARGOS-RAL在不同噪声水平和样本量下识别经典PDEs的表现，证明了其在处理含噪声和非均匀分布数据时的鲁棒性。我们还测试了该算法在仅由随机噪声构成的数据集上的性能，以模拟数据质量严重受损的场景。结果表明，ARGOS-RAL能够有效且可靠地从数据中识别潜在PDEs，在大多数情况下优于序贯阈值岭回归方法。我们强调了结合统计方法、机器学习和动力系统理论从采集数据中自动发现控制方程的潜力，这将推动科学建模过程的简化。