<p dir="ltr">Data is collected everywhere in our world; however, it often is noisy and incomplete. Different sources of data may have different characteristics, quality levels, or come from dynamic and diverse environments. This poses challenges for both humans who want to gain insights from data and machines which are learning patterns from data. How can we leverage the diversity of data across domains to enhance our understanding and decision-making? In this thesis, we address this question by proposing novel methods and applications that use multiple domains as more holistic sources of information for both human and machine learning tasks. For example, to help human operators understand environmental dynamics, we show the detection and localization of distribution shifts to problematic features, as well as how interpretable distributional mappings can be used to explain the differences between shifted distributions. For robustifying machine learning, we propose a causal-inspired method to find latent factors that are robust to environmental changes and can be used for counterfactual generation or domain-independent training; we propose a domain generalization framework that allows for fast and scalable models that are robust to distribution shift; and we introduce a new dataset based on human matches in StarCraft II that exhibits complex and shifting multi-agent behaviors. We showcase our methods across various domains such as healthcare, natural language processing (NLP), computer vision (CV), etc. to demonstrate that learning from data across domains can lead to more faithful representations of data and its generating environments for both humans and machines.</p>

<p dir="ltr">Neural operator learning is emerging as a prominent technique in scientific machine learn- ing for modeling complex nonlinear systems with multi-physics and multi-scale applications. A common drawback of such operators is that they are data-hungry and the results are highly dependent on the quality and quantity of the training data provided to the models. Moreover, obtaining high-quality data in sufficient quantity can be computationally prohibitive. Faster surrogate models are required to overcome this drawback which can be learned from datasets of variable fidelity and also quantify the uncertainty. In this work, we propose a Multi-Level Stacked Deep Operator Network (MLSDON) which can learn from datasets of different fidelity and is not dependent on the input function. Through various experiments, we demonstrate that the MLSDON can approximate the high-fidelity solution operator with better accuracy compared to a Vanilla DeepONet when sufficient high-fidelity data is unavailable. We also extend MLSDON to build robust confidence intervals by making conformalized predictions. This technique guarantees trajectory coverage of the predictions irrespective of the input distribution. Various numerical experiments are conducted to demonstrate the applicability of MLSDON to multi-fidelity, multi-scale, and multi-physics problems.</p>