  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.
