Data-driven discovery of partial differential equations (PDEs) has emerged as a promising approach for deriving governing physics when domain knowledge about observed data is limited. Despite recent progress, the identification of governing equations and their parametric dependencies using conventional information criteria remains challenging in noisy situations, as the criteria tend to select overly complex PDEs. In this paper, we introduce an extension of the uncertainty-penalized Bayesian information criterion (UBIC), which is adapted to solve parametric PDE discovery problems efficiently without requiring computationally expensive PDE simulations. This extended UBIC uses quantified PDE uncertainty over different temporal or spatial points to prevent overfitting in model selection. The UBIC is computed with data transformation based on power spectral densities to discover the governing parametric PDE that truly captures qualitative features in frequency space with a few significant terms and their parametric dependencies (i.e., the varying PDE coefficients), evaluated with confidence intervals. Numerical experiments on canonical PDEs demonstrate that our extended UBIC can identify the true number of terms and their varying coefficients accurately, even in the presence of noise. The code is available at \url{https://github.com/Pongpisit-Thanasutives/parametric-discovery}.

翻译：数据驱动的偏微分方程发现已成为在观测数据领域知识有限时推导主导物理规律的有效方法。尽管近期取得进展，但在噪声环境下，使用传统信息准则识别主导方程及其参数依赖性仍面临挑战，因为这些准则倾向于选择过度复杂的偏微分方程。本文提出了不确定性惩罚贝叶斯信息准则的扩展版本，该改进方法适用于高效求解参数化偏微分方程发现问题，且无需计算代价高昂的偏微分方程数值模拟。该扩展准则通过量化不同时空点上的偏微分方程不确定性来防止模型选择中的过拟合现象。基于功率谱密度的数据变换计算该准则，可发现真正捕捉频域空间定性特征的主导参数化偏微分方程，其仅包含少量显著项及其参数依赖性（即变化的偏微分方程系数），并通过置信区间进行评估。经典偏微分方程的数值实验表明，即使在噪声存在的情况下，我们扩展的不确定性惩罚贝叶斯信息准则仍能准确识别真实项数及其变化系数。代码发布于 \url{https://github.com/Pongpisit-Thanasutives/parametric-discovery}。