PDE-Constrained Machine Learning with Gaussian Processes towards Digital Twins

The development of trustworthy digital twins for complex physical systems, often governed by high-dimensional partial differential equations (PDEs), demands predictive models that are both highly accurate and provide reliable uncertainty quantification (UQ). This thesis addresses the complementary weaknesses of two leading machine learning paradigms: the powerful expressiveness of deep neural networks (NNs), which often lack robust UQ, and the principled probabilistic nature of Gaussian processes (GPs), which struggle with the curse of dimensionality.

This work presents a unified framework that synergizes these approaches through physics-constrained deep kernel learning (DKL). In this hybrid architecture, an NN learns a low-dimensional, feature representation, which in turn defines the kernel of a GP. This design leverages the representation learning power of NNs to make GP inference tractable and effective in high-dimensional settings.

The core contributions are twofold. For forward problems, we introduce a PDE-constrained DKL model that effectively mitigates the curse of dimensionality. By embedding physical laws into deep kernel, the framework learns physically consistent solutions and provides reliable uncertainty estimates even from sparse data. For inverse problems, such as estimating unknown PDE parameters, we propose a novel two-stage Bayesian inference strategy to overcome computational intractability. An initial physics-informed pretraining stage optimizes the high-dimensional NN weights and provides robust point estimates. In the second stage, these weights are fixed, enabling efficient Hamiltonian Monte Carlo (HMC) sampling of the posterior distribution for only the low-dimensional PDE parameters and kernel hyperparameters.

Collectively, this dissertation delivers a cohesive and robust framework for scientific machine learning. By systematically addressing architectural foundations, forward modeling in high dimensions, and tractable Bayesian inference for inverse problems, this work provides a significant step towards building trustworthy, uncertainty-aware digital twins for complex systems in science and engineering.