Scientific Machine Learning for Modeling and Discovery of Physical Systems with Quantified Uncertainty

Deep learning models have nowadays gained increasing attention in the scientific computing field due to their inherent nature to capture nonlinear and high-dimensional features. The main motivation of my research is to leverage physics laws and data to develop predictive deep learning model for physics systems. On the one hand, when the physics laws is known but the data is little, the deep learning model can be used to develop surrogate modeling of the high-accuracy simulation or measurements instead of directly replacing the canonical models. On the other hand, when the data is abundant but the underlying physics is partly known, equation discovery model can be used to identify the underlying governing equations. Under both circumstances, it is of great importance to quantify the uncertainty of the developed models. This dissertation demonstrates that deep learning based model is a good candidate for both surrogate modeling and equation discovery from several perspectives. Firstly, we developed a data-free physics-informed surrogate modeling of fluid flows without relying on any simulation data. Secondly, when the flow measurements are sparse and noisy, we propose an innovative physics-constrained Bayesian deep learning approach to reconstruct flow fields from corrupted velocity data. Several test cases on idealized vascular flows with synthetic measurement data are studied to demonstrate the merit of the proposed method. In the third part, we aim to discover the governing equations for dynamical systems from limited observation data. To achieve this goal, we first developed a deep sparse Bayesian learning approach to uncover the explicit governing equation of a parametric dynamic system, where predictive uncertainty will be estimated. To further improve the robustness against data noise, we propose another Bayesian spline learning approach for equation discovery. Several linear/nonlinear ODE/PDE systems are used to demonstrate the effectiveness. In the last part, we also proposed to use a normalizing flow based model to predict complex dynamics. The preliminary result shows that the prediction is accurate and the model can give reasonable forward uncertainty estimation.