Efficient scientific machine learning algorithms for surrogate modeling and inverse problems
Creators: name: Lu, Chuan
Publisher: name: University of Iowa
Year: 2022
Subjects: Inverse Problem
MRI Fingerprinting
Multi-Fidelity Modeling
Physics-Informed Neural Networks
Scientific Machine Learning
Uncertainty Quantification
public abstract

description: In this thesis, we present several approaches to scientific machine learning algorithms and their applications in surrogate modeling and inverse problems. Specifically, it addresses the following topics:
description: 1. Feature-augmented approach. We propose a nonintrusive reduced basis method when a cheap low-fidelity model and an expensive high-fidelity model are available. The method employs the proper orthogonal decomposition method (POD) to generate the high-fidelity reduced basis and a neural network to learn the high-fidelity reduced coefficients. In contrast to previously proposed methods, besides the model parameters, we also augmented the features extracted from the data generated by an efficient bi-fidelity surrogate as the input feature of the proposed neural network. By incorporating relevant bi-fidelity features, we demonstrate that such an approach can improve the predictive capability and robustness of the neural network via several benchmark examples. Due to its nonintrusive nature, it is also applicable to general parameterized problems.
description: 2. Physics-informed approach. The SIR (Susceptible-Infected-Removed) compartmental epidemic spread model has been widely used for mathematical modeling of epidemics. In the SIR model, the contact rate and recovery rate are among the most important characteristic parameters that control the evolution of an epidemic. Accurate inference of these parameters from measurement data is essential for understanding the principles of epidemic spreads and choosing a proper policy for disease control. We apply the physics-informed neural network (PINN) framework to solve inverse problems based on the SIR model. PINN was first developed in 2017 and has since gained widespread adoption in scientific and engineering applications. PINN embeds equation information into neural networks by minimizing equation residual loss. We demonstrate through numerical examples based on the SIR model, that with partial observations of state variables, and with scattered and noisy measurements, PINN can accurately infer unknown physical parameters and jointly produce predictions for the forward problem.
description: 3. Physics-based approach. Magnetic resonance imaging (MRI) fingerprinting is a versatile technique in medical imaging for recovering exact values of relaxation rates for tissues. We propose a novel method for MRI fingerprinting based on Bloch equations, the physical principles of MRI fingerprinting. Specifically, we follow a Bayesian approach and employ a flow-based generative model to approximate the posterior distribution of target parameters and model signal in frequency space using the physics-based forward operator. We demonstrate the capability of the proposed physics-based generative model in approximating target parameters accurately as well as providing meaningful uncertainty quantification with both noise-free and noisy measurements of an anatomic dataset.