A Study of Physics-Informed Deep Learning for System Fluid Dynamics Closures
The sub-grid-scale (SGS) physics models, or so-called closure relations (CR), are essential in thermal-fluid modeling and simulation codes, ranging from large-eddy simulation to system-level thermo-fluid dynamic problems. While details vary, all thermo-fluid codes solve coupled mass-momentum-energy conservation partial differential equations (PDE). Traditionally, the analytical form of closure models has to be known, and then we calibrate the model using data from a set of applicable experiments. Characteristically, it takes long effort to gain insights and mechanistic understanding needed for developing closure models. Such a model is a compact form of data. The long time required for new model development limits the pace of model application while dealing with new geometries, new coolants or system conditions, particularly in newly designed systems. Alternative to mechanistic and semi-analytical models, it is possible to discover the pattern of closure relations by applying the modern machine learning methods to the body of relevant experimental data. Deep learning (DL) is a relatively new branch in the machine learning family, and a major breakthrough has been achieved since Hinton introduces a fast algorithm to efficiently train a feed-forward neural network (NN) with multilayer perceptrons. Generally, any NN with more than two layers is considered as deep learning, and it contains a large number of hyper-parameters to achieve pattern recognition or regression for complex datasets. Hornik shows that multilayer NN is compatible with universal approximation theorem. There is no constraint for a multilayer NN to capture the properties of any measurable information. This capability makes DL attractive for closure models in a system thermal-hydraulic code. As system geometry and/or operating conditions change, new physical experiments (increasingly sophisticated) are conducted to support evaluation and revision of closure models. Such a so-called data-driven modeling relaxes the structural form of system codes to directly assimilate data instead of hard-wiring the model to pre-calibrated correlations. The objective of this research is to enable cost-effective development of a DL-based closure that reflects the data. The previous work on applying machine learning techniques to fluid dynamics focused on statistical learning with basic NN from direct numerical simulations (DNS) dataset. In the present work, we show that DL-based closure models can cause the fluid calculation to diverge if multilayer NN is used. To address the issue, we investigate conditions by which the DL-based closure models work compatibly, stably, and effectively with PDE-constrained forward prediction problems. Optimally meeting the compatibility conditions constitutes a physics-informed deep learning (PIDL) strategy. We refer to this process as PDE-constrained machine learning or PDE-ML. To develop the technical basis for PDE-ML using DL-based fluid dynamics closure, numerical experiments are performed for manufactured problems of increasing complexity. (authors)