Robust physics (e.g., governing equations and laws) discovery is of great interest for many engineering fields and explainable machine learning. A critical challenge compared with general training is that the term and format of governing equations are not known as a prior. In addition, significant measurement noise and complex algorithm hyperparameter tuning usually reduces the robustness of existing methods. A robust data-driven method is proposed in this study for identifying the governing Partial Differential Equations (PDEs) of a given system from noisy data. The proposed method is based on the concept of Progressive Sparse Identification of PDEs (PSI-PDE or ψ-PDE). Special focus is on the handling of data with huge uncertainties (e.g., 50% noise level). Neural Network modeling and fast Fourier transform (FFT) are implemented to reduce the influence of noise in sparse regression. Following this, candidate terms from the prescribed library are progressively selected and added to the learned PDEs, which automatically promotes parsimony with respect to the number of terms in PDEs as well as their complexity. Next, the significance of each learned terms is further evaluated and the coefficients of PDE terms are optimized by minimizing the L2 residuals. Results of numerical case studies indicate that the governing PDEs of many canonical dynamical systems can be correctly identified using the proposed ψ-PDE method with highly noisy data. Codes of all demonstrated examples are available on the website: https://github.com/ymlasu. One great benefit of proposed algorithm is that it avoids complex algorithm modification and hyperparameter tuning in most existing methods. Limitations of the proposed method and major findings are presented.

稳健的物理学（例如，控制方程和定律）发现对许多工程领域和可解释的机器学习都非常感兴趣。与一般培训相比，一个关键的挑战是控制方程的术语和格式不是先验的。此外，显着的测量噪声和复杂的算法超参数调整通常会降低现有方法的鲁棒性。本研究提出了一种稳健的数据驱动方法，用于从噪声数据中识别给定系统的控制偏微分方程 (PDE)。所提出的方法基于 PDE 的渐进稀疏识别的概念（PSI-PDE 或ψ-PDE）。特别关注具有巨大不确定性（例如，50% 的噪声水平）的数据的处理。实施神经网络建模和快速傅立叶变换 (FFT) 以减少稀疏回归中噪声的影响。在此之后，从规定的库中逐步选择候选术语并将其添加到学习的 PDE 中，这会自动促进 PDE 中术语数量及其复杂性的简约性。接下来，进一步评估每个学习项的重要性，并通过最小化 L2 残差来优化 PDE 项的系数。数值案例研究的结果表明，可以使用建议的ψ正确识别许多典型动力系统的控制偏微分方程-具有高噪声数据的 PDE 方法。网站上提供了所有演示示例的代码：https://github.com/ymlasu。所提出算法的一大好处是它避免了大多数现有方法中复杂的算法修改和超参数调整。提出了所提出的方法和主要发现的局限性。