Physics-informed machine learning methods for environmental modeling and uncertainty quantification

Machine learning approaches are increasingly recognized for enhancing prediction and model inversion in natural and engineering systems due to their high expressivity, improved training algorithms, and fast inference speed. However, effectively applying data-driven machine learning methods for environmental problems, particularly in data-limited contexts such as subsurface modeling and large-scale climate simulations, remains challenging. One major limitation is the scarcity of high-quality data that accurately captures the heterogeneous and multi-scale nature of these systems. Additionally, environmental problems are often high-dimensional, making them susceptible to the ``curse of dimensionality (CoD)." This poses significant computational challenges, especially in inverse parameter estimation, uncertainty quantification (UQ), and design and optimization tasks, where solving these problems requires extensive physics-based simulations. To address these challenges, our research aims to develop machine learning approaches that integrate physical domain knowledge (i.e., governing physical equations) as a form of regularization. Additionally, we explore efficient algorithms to mitigate the effects of high dimensionality in inverse problems and UQ. Our research unfolds into three well-defined objectives: The first is physics-informed machine learning for deterministic PDE problems. Here, we focus on developing, improving, and analyzing physics-informed machine learning approaches in the deterministic regime, designed to accurately approximate solutions to partial differential equations (PDEs) as well as learning PDE operators. We propose an improved training framework for physics-informed neural networks (PINNs) that systematically targets different sources of PINN errors. The effectiveness of our improved PINN model is demonstrated in forward, inverse, and backward advection-dispersion equations (ADEs) with sharply perturbed initial conditions, where conventional PINNs struggle to learn effectively. We also introduce a reduced-order method called the physics-informed Karhunen-Lo\`eve expansion (PICKLE), which combines model reduction techniques with physics-informed machine learning to efficiently solve space-time-dependent PDE problems. A key advantage of these two methods is their ability to seamlessly assimilate additional data, which can enhance model training and improve predictive accuracy. Additionally, we propose reduced-order neural operator surrogate models, including KL-DNN and VAE-DNN, which employ the dimension reduction technique to identify low-dimensional subspaces and construct operator-based mappings between PDE parameters and solutions. These surrogate models offer significant computational and energy efficiency, as they can be trained in separate components while maintaining accuracy comparable to state-of-the-art neural operators such as the Fourier Neural Operator. The second objective is to develop scalable UQ methods to address the challenges posed by complex posterior distributions in high-dimensional, PDE-constrained inverse problems. These problems are often affected by the CoD and exhibit pathological behaviors in high-dimensional probability spaces, making existing Bayesian inference computationally demanding. We propose a Bayesian UQ framework based on the randomize-then-optimize approach, which enables efficient posterior sampling by tailoring the optimization process to the loss functions of our physics-informed machine learning models, including PINN and PICKLE. The third objective is to develop effective digital twins capable of accurately and rapidly predicting PDE solutions under a wide range of control variables while remaining differentiable with respect to them. To achieve this, we integrate the proposed neural operator-based surrogate models with transfer learning techniques, allowing models trained on one set of conditions to generalize efficiently to new scenarios with minimal retraining. Additionally, we provide a rigorous mathematical analysis of the transferability of these surrogate models for both linear and nonlinear PDEs. This thesis outlines a comprehensive roadmap for advancing physics-informed machine learning techniques. We present numerical results for each objective, demonstrating their effectiveness and laying the foundation for broader applications toward large, real-world problems.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Civil Engineering

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Zong, Yifei

Contributors dc:contributor

• Tartakovsky, Alexandre
• Valocchi, Albert
• Kumar, Praveen
• Meidani, Hadi

• Physics-informed Machine Learning
• Environmental Modeling
• Uncertainty Quantification

Statement dc:rights

• Copyright 2025 Yifei Zong

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129437

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129437