不确定性下稳健的数据驱动偏微分方程发现
Robust Data-Driven Discovery of Partial Differential Equations under Uncertainties
摘要：
强大的物理学（例如，控制方程和定律）的发现对于许多工程领域和可解释的机器学习都是非常有意义的。与一般训练相比，一个关键的挑战是，控制方程的术语和格式不被认为是先验的。此外，显著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。提出了一种鲁棒的数据驱动方法，用于从噪声数据中识别给定系统的控制偏微分方程（PDE）。所提出的方法是基于PDE的渐进稀疏识别（PSI-PDE或$ψ$ -PDE）的概念。特别关注具有巨大不确定性的数据的处理（例如，50 $\%$噪声级）。采用神经网络建模和快速傅里叶变换（FFT）来降低稀疏回归中噪声的影响。在此之后，从指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。接着，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。数值算例的结果表明，许多正则动力系统的控制偏微分方程可以正确识别使用建议的$ψ$ -PDE方法与高噪声数据。该算法的一个优点是避免了现有方法中复杂的算法修改和超参数调整。所提出的方法和主要研究结果的局限性。
英文摘要：
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 is 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. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
