Generative Artificial Intelligence and Uncertainty Quantification for Scientific Discovery

Scientific machine learning models can achieve remarkable accuracy, yet fail silently when data are scarce, predictions extend beyond the training distribu- tion, or errors accumulate during deployment. Together these limit their use in settings where an overconfident prediction could yield an unphysical result or misguide an expensive downstream experiment. This thesis investigates whether scientific machine learning models can be designed to recognise what they do not know, while remaining computationally efficient, scalable, and faithful to the physical structure of the underlying problem. To address this question, this thesis develops efficient and scalable vari- ational Bayesian methods across multiple modalities of scientific machine learning. These encompass causal dynamical systems, geometrically struc- tured models subject to invariance and equivariance constraints, and large pretrained models adapted through parameter-efficient fine-tuning. Although these settings differ in their structure and computational requirements, they share the need to represent uncertainty without compromising the inductive biases that make scientific models effective. The results demonstrate that principled Bayesian uncertainty need not re- quire either prohibitively expensive posterior sampling or the loss of domain- specific structure. On the contrary, we show that it is possible to develop structured and computationally tractable Bayesian models for scientific ma- chine learning. By treating uncertainty as a first-class component of scientific machine learning, this thesis lays the groundwork for a future generation of uncertainty-aware simulation, active learning pipelines, and scientific decision- making at scale.