We propose a new parameter-adaptive uncertainty-penalized Bayesian information criterion (UBIC) to prioritize the parsimonious partial differential equation (PDE) that sufficiently governs noisy spatial-temporal observed data with few reliable terms. Since the naive use of the BIC for model selection has been known to yield an undesirable overfitted PDE, the UBIC penalizes the found PDE not only by its complexity but also the quantified uncertainty, derived from the model supports' coefficient of variation in a probabilistic view. We also introduce physics-informed neural network learning as a simulation-based approach to further validate the selected PDE flexibly against the other discovered PDE. Numerical results affirm the successful application of the UBIC in identifying the true governing PDE. Additionally, we reveal an interesting effect of denoising the observed data on improving the trade-off between the BIC score and model complexity. Code is available at https://github.com/Pongpisit-Thanasutives/UBIC.

翻译：我们提出了一种新的参数自适应不确定性惩罚贝叶斯信息准则（UBIC），以优先选择能够用少量可靠项充分解释含噪时空观测数据的简约偏微分方程（PDE）。由于已知直接使用BIC进行模型选择会产生不理想的过拟合PDE，UBIC不仅根据其复杂性对找到的PDE进行惩罚，还根据从概率视角下模型支持变量的变异系数导出的量化不确定性进行惩罚。我们还引入了物理信息神经网络学习作为一种基于模拟的方法，以灵活地验证所选PDE相对于其他已发现PDE的有效性。数值结果证实了UBIC在识别真实控制PDE中的成功应用。此外，我们揭示了去噪观测数据对改善BIC得分与模型复杂性之间权衡的有趣效果。代码可在https://github.com/Pongpisit-Thanasutives/UBIC获取。