Reduced-Complexity Modeling of Multi-Scale Problems in the Low Data Regime

Many-query scientific and industrial applications, such as design, demand affordable yet accurate computational models. Data-driven Reduced-Complexity Models (RCMs), which are the focus of this thesis, typically require a significant amount of training data. Despite efficiency gains in the prediction stage, the large expense in the generation of the training data contradicts the low-cost pursuit of a RCM, and can make the {em creation} of the RCM unaffordable. With this motivation, this thesis proposes a number of techniques towards the end of addressing the development of RCMs in what we refer to as the {em low data regime}, which can manifest itself in three ways in practical problems: 1) Spatially, high-fidelity simulations can only be conducted on a truncated sub-domain, or for an individual component of a larger system; 2) Temporally, only a short period of dynamics can be simulated, spanned by a limited number of snapshots; and 3) Parameterically, only a limited set of operating conditions can be covered in training. As a motivating example, this thesis considers a rocket engine combustor, for which a direct full-system simulation is not affordable. In concert with the underlying physics and design requirements, the computational domain is decomposed into 1) individual injectors and their neighboring downstream regions, which share a few types of identical geometries, yet experience computationally expensive reacting flow dynamics; and 2) the further downstream chamber and nozzle, which experiences relatively simple acoustics-dominated dynamics, yet can be connected with different numbers and arrangements of injectors. The responses to different upstream configurations, and possible consequential changes in the computational grid and even the geometry cannot be addressed by current data-driven RCMs. Our approach uses a RCM for the former, and an inexpensive PDE solver for the latter, such that the efficiency of a RCM and the flexibility of a PDE solver are utilized at the same time. Ideally, the training of the RCM only depends on independently-generated solutions for a single injector region, and no full-system full-order simulation is required. The realization of the strategy depends on three key aspects: 1) reduced-domain training -- training data generation in an isolated reduced-domain, 2) model design -- design of efficient and predictive RCMs, and 3) model-solver coupling -- effective interfacing methods to couple a data-driven RCM and a high-fidelity solver. The data-driven RCMs proposed in this work are not restricted to specific applications, except for the need to adapt the inputs and outputs. For predictions on a fixed geometry, non-intrusive ROMs provide remarkable efficiency improvements, especially when the time period of interest is also fixed across different parameters such that the multi-level autoencoder network is viable. Local surrogate models eliminate the dependence on the global spatio-temporal domain, and generalize to more applications. With the help of interface models, the model-solver coupling approaches are also shown to generalize to different types and schemes of models and solvers. The resulting multi-domain framework endows significantly more flexibility to the applications of ROMs, yet still requires a fixed geometry in the ROM sub-domain. For more generalizable models such as the CP-GNet, the framework also reduces the training effort and improves the prediction performance. Despite the encouraging results, error estimation and stability properties require formal quantification, and remains an open topic for further research.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/172659/1/davidxu_1.pd