Uncertainty Quantification in Scientific Machine Learning: Theory, Methods, and SoftwaredissertationsZou, ZongrencreatorKarniadakis, GeorgeAdvisorDarbon, JeromeReaderOwhadi, HoumanReaderTartakovsky, DanielReaderBrown University. Department of Applied Mathematicssponsor2024, None p.born digitalThesis (Ph. D.)--Brown University, 2024thesesUncertainty quantification (UQ) is a critical component in scientific machine learning (SciML), addressing the challenges of modeling, predicting, and interpreting complex systems under incomplete or imprecise data or models. This dissertation presents a comprehensive study on the theory, methods, and software for UQ in SciML, focusing on both theoretical advancements and practical implementations. We begin by addressing the challenges of data sparsity and high noise levels through the development of multi-head physics-informed neural networks (MH-PINNs), which leverage functional priors in solving ordinary/partial differential equations (ODEs/PDEs) with UQ. Next, a novel theoretical connection between Bayesian inference and Hamilton-Jacobi (HJ) PDEs is introduced, enabling efficient and interpretable methodologies for large-scale data problems, incremental/continual learning and hyperparameter tuning. The impact of noise model misspecifications is further examined through the development of a Bayesian approach for quantifying uncertainty arising from noisy inputs and outputs in physics-informed neural networks (PINNs) and neural operators (NOs). This equips these methods to handle noisy inputs and outputs effectively. Additionally, the dissertation addresses physical model misspecifications by presenting a framework that integrates auxiliary neural networks (NNs) to correct and quantify uncertainties in flawed models. To bridge the gap between theory, methodology, and practical implementation, the dissertation introduces NeuralUQ, an open-source software library for UQ in SciML. NeuralUQ provides a flexible, user-friendly platform designed to support researchers and practitioners in their work.Applied mathematicsEnglishRPB20250310