Scientific Machine Learning is a new class of approaches that integrate physical knowledge and mechanistic models with data-driven techniques for uncovering governing equations of complex processes. Among the available approaches, Universal Differential Equations (UDEs) are used to combine prior knowledge in the form of mechanistic formulations with universal function approximators, like neural networks. Integral to the efficacy of UDEs is the joint estimation of parameters within mechanistic formulations and the universal function approximators using empirical data. The robustness and applicability of resultant models, however, hinge upon the rigorous quantification of uncertainties associated with these parameters, as well as the predictive capabilities of the overall model or its constituent components. With this work, we provide a formalisation of uncertainty quantification (UQ) for UDEs and investigate important frequentist and Bayesian methods. By analysing three synthetic examples of varying complexity, we evaluate the validity and efficiency of ensembles, variational inference and Markov chain Monte Carlo sampling as epistemic UQ methods for UDEs.

翻译：科学机器学习是一类新兴方法，它将物理知识与机理模型同数据驱动技术相结合，用于揭示复杂过程的控制方程。在现有方法中，通用微分方程（UDEs）被用于将机理形式的先验知识与通用函数逼近器（如神经网络）相结合。UDEs有效性的关键在于利用经验数据对机理公式中的参数与通用函数逼近器进行联合估计。然而，所得模型的稳健性和适用性取决于对这些参数相关不确定性的严格量化，以及整体模型或其组成部分的预测能力。本研究提出了通用微分方程不确定性量化（UQ）的形式化框架，并探究了重要的频率学派与贝叶斯方法。通过分析三个不同复杂度的合成示例，我们评估了集成方法、变分推断和马尔可夫链蒙特卡洛采样作为UDEs认知不确定性量化方法的有效性与效率。