A paradigm for data-driven predictive modeling using field inversion and machine learning

Abstract

We propose a modeling paradigm, termed field inversion and machine learning (FIML), that seeks to comprehensively harness data from sources such as high-fidelity simulations and experiments to aid the creation of improved closure models for computational physics applications. In contrast to inferring model parameters, this work uses inverse modeling to obtain corrective, spatially distributed functional terms, offering a route to directly address model-form errors. Once the inference has been performed over a number of problems that are representative of the deficient physics in the closure model, machine learning techniques are used to reconstruct the model corrections in terms of variables that appear in the closure model. These reconstructed functional forms are then used to augment the closure model in a predictive computational setting. As a first demonstrative example, a scalar ordinary differential equation is considered, wherein the model equation has missing and deficient terms. Following this, the methodology is extended to the prediction of turbulent channel flow. In both of these applications, the approach is demonstrated to be able to successfully reconstruct functional corrections and yield accurate predictive solutions while providing a measure of model form uncertainties.

Machine Learning for Fluid Mechanics

“…Machine learning has been used to identify and model discrepancies in the Reynolds stress tensor between a RANS model and high-fidelity simulations (Ling & Templeton 2015;Parish & Duraisamy 2016;Ling et al 2016b;Xiao et al 2016;Singh et al 2017;. Ling & Templeton (2015) compare support vector machines, Adaboost decision trees, and random forests to classify and predict regions of high uncertainty in the Reynolds stress tensor.…”

“…Xiao et al (2016) leveraged sparse online velocity measurements in a Bayesian framework to infer these discrepancies. In related work, Parish & Duraisamy (2016) develop the field inversion and machine learning modeling framework, that builds corrective models based on inverse modeling. This framework was later used by Singh et al (2017) to develop a neural network enhanced correction to the Spalart-Allmaras RANS model, with excellent performance.…”

Coercing machine learning to output physically accurate results

“…Since a number of authors have begun to consider the use of machine/deep learning for problems in traditional computational physics, see e.g. [1,2,3,4,5,6,7,8,9,10,11,12], we are motivated to consider methodologies that constrain the interpolatory results of a network to be contained within a physically admissible region. Quite recently, [13] proposed adding physical constraints to generative adversarial networks (GANs) also considering projection as we do, while stressing the interplay between scientific computing and machine learning; we refer the interested reader to their work for even more motivation for such approaches.…”

Data-Driven Reduced Model Construction with Time-Domain Loewner Models

“…In [39,48], corrections to reduced models are inferred with Bayesian inference for several different parameter configurations. The inferred corrections with the corresponding parameter configurations are used as a training set to learn a map from the parameters of the model to the corrections with supervised machine learning techniques.…”

“…The inferred corrections with the corresponding parameter configurations are used as a training set to learn a map from the parameters of the model to the corrections with supervised machine learning techniques. The inference and learning approach presented in [39,48] is demonstrated on applications in the context of model reduction for turbulent flow models. The works [31,52] present a data assimilation framework for correcting the model bias of reduced models with data.…”