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.

科学机器学习是一类新的方法，它将物理知识和机械模型与数据驱动技术相结合，以揭示复杂过程的控制方程。本文提供了不确定性量化 (UQ) 的 UDE 的形式化，并研究了重要的频率派和贝叶斯方法。通过分析三个不同复杂度的合成示例，本文评估了集成、变分推断和马尔可夫链蒙特卡洛采样作为 UDE 的认识论不确定性量化方法的有效性和效率。