Data-driven uncertainty quantification in ordinary and partial differential equation models

Abstract

This doctoral dissertation focuses on developing two data-driven methods to quantify uncertainty inordinary and partial differential equation models. The first method addresses the case where the uncertainty is characterized by finite-dimensional randomvariables. In this case, polynomial chaos hasbeen proven to be an efficient method for uncertainty quantification, requiring much smaller simulations than the Monte Carlo, a traditional sample-based method. However, its applicability is limitedby the rapid decrease in efficiency for high-dimensional problems, the requirement of exact knowledge about the probability distribution, and the assumption of mutual independence of the random variables. We describe a new data-driven method to overcome these limitations. It is based on the transformation of correlated random variables into independent random variables. We use singular value decomposition as a transformation strategy that does not require information about the probability distribution. For the transformed random variables, we can construct the polynomial chaos basis to build the approximate solution. This approach provides an additional benefit of quantifying high-dimensional uncertainty by combining our method with the analysis-of-variance (ANOVA) method. We demonstrate in several numerical examples that our proposed approach leads to accurate solutions with a much smaller number of simulations compared to the Monte Carlo method.

Subsequently, we turn our attention to the scenario where the uncertainty arises from infinite-dimensional random parameters measured bya limited number of sensors. We propose a novel data-driven Bayesian deep learning method for uncertainty quantification in this case. By modeling the solution using a Bayesian neuralnetwork, we seamlessly integrate the underlying physical laws into the posterior distribution of the network parameters through automatic differentiation. To effectively learn the posterior distribution of numerous parameters, we employ Hamiltonian MonteCarlo, an efficient method for high-dimensional sampling. A parallelization of our algorithm mitigates the high computational cost associated with Hamiltonian Monte Carlo, thereby enhancing its efficiency. We provide various numerical examples to demonstrate theeffectiveness of our proposed method for uncertainty quantification in both forward and inverse problems. Furthermore, we present promising results indicating that the computational cost is nearly independent of the random dimension of the problem, demonstrating the method's potential for tackling the so-called curse of dimensionality.