Deep Learning For Surrogate Modeling And Uncertainty Quantification In Science & Engineering

Scientific machine learning (SciML) has become an increasingly important tool for constructing surrogate models of complex physical systems, enabling rapid approximation of expensive numerical solvers and supporting tasks such as design optimization, uncertainty analysis, and autonomous experimentation. However, scientific surrogates are often deployed in data-scarce and extrapolative regimes, where predictive accuracy alone is insufficient. Reliable uncertainty quantification and appropriate inductive biases are essential for ensuring that model predictions remain trustworthy and useful for downstream decision-making. This thesis first develops uncertainty-aware surrogate models that distinguish between epistemic and aleatoric uncertainty. To address epistemic uncertainty in operator learning, it introduces Neural Epistemic Operator Networks (NEON), which integrate the Epistemic Neural Network framework with neural operators for learning function-to-function mappings. This design enables scalable and well-calibrated epistemic uncertainty estimates using a single operator network, and is leveraged in composite Bayesian optimization problems over function spaces to improve sample efficiency. The thesis then focuses on aleatoric uncertainty arising from intrinsic non-uniqueness in scientific problems, such as ill-posed inverse mappings, multistability, and chaotic dynamics. In these settings, standard regression trained with mean squared error collapses multimodal solution sets to conditional averages that can be physically invalid. To address this limitation, the thesis revisits Mixture Density Networks as explicit probabilistic models for multimodal conditional distributions, demonstrating improved data efficiency, interpretability, and mode recovery in representative scientific benchmarks. Finally, the thesis turns to the complementary challenge of architectural inductive bias in scientific surrogate modeling, with a focus on physics-informed neural networks (PINNs). This setting represents a regime in which models may be trained with little or no observational data and must rely almost entirely on their ability to represent functions and their derivatives accurately in order to satisfy differential constraints. Motivated by approximation theory, it introduces ActNet, a scalable architecture inspired by modern variants of the Kolmogorov Superposition Theorem. ActNet provides theoretical guarantees on function approximation, stable activation scaling, and derivative expressivity, offering a principled alternative to existing Kolmogorov--Arnold Network formulations and to generic multilayer perceptrons in physics-informed learning. Together, these contributions advance uncertainty-aware and structure-aware surrogate modeling as a foundation for reliable and data-efficient scientific computation.

Author dc:creator

• Ferreira Guilhoto, Leonardo

Advisor dc:contributor.advisor

• Perdikaris, Paris

• Mathematics
• Computer Sciences
• Physics

Language dc:language.iso

en

Repository record dc:identifier.uri

https://repository.upenn.edu/handle/20.500.14332/62712

OAI identifier oai:identifier

oai:repository.upenn.edu:20.500.14332/62712