Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

Geneva, Nicholas; Zabaras, Nicholas

doi:10.1016/j.jcp.2019.01.021

Cited by 110 publications

(42 citation statements)

References 46 publications

(79 reference statements)

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“…Recently, data-driven turbulence modeling has emerged as a promising field of research with a number of approaches been proposed in the past few years (e.g., Ling et al 2016;Singh & Duraisamy 2016;Wang et al 2017;Weatheritt & Sandberg 2016). Of particular relevance to the present discussions are the data-driven Reynolds stress closures (Ling et al 2016;Wang et al 2017;Geneva & Zabaras 2018) in which the Reynolds stresses are obtained directly from machine learning models trained on high-fidelity simulation databases without solving partial differential equations (PDEs) or using explicit algebraic models. In such models, the chain coupling mechanism related to the Reynolds stress transport equations as described above is not a relevant cause of numerical instability, and thus the possible ill-conditioning of RANS equations is placed under spotlight.…”

Section: Unique Challenges In Data-driven Reynolds Stress Closuresmentioning

confidence: 99%

Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

et al. 2019

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Reynolds-averaged Navier-Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy viscosity models are intrinsically unable to handle flows with nonequilibrium turbulence (e.g., flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5% from direct numerical simulation (DNS) databases into RANS equations leads to velocities with large errors (up to 35%). While such an observation may have only marginal relevance to traditional Reynolds stress models, it is disturbing for the recently emerging data-driven models that treat the Reynolds stress as an explicit source term in the RANS equations, as it suggests that the RANS equations with such models can be ill-conditioned. So far, a rigorous analysis of the condition of such models is still lacking. As such, in this work we propose a metric based on local condition number function for a priori evaluation of the conditioning of the RANS equations. We further show that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. Comprehensive numerical tests are performed on turbulent channel flows at various Reynolds numbers and additionally on two complex flows, i.e., flow over periodic hills and flow in a square duct. Results suggest that the proposed metric can adequately explain observations in previous studies, i.e., deteriorated model conditioning with increasing Reynolds number and better conditioning of the implicit treatment of Reynolds stress compared to the explicit treatment. This metric can play critical roles in the future development of data-driven turbulence models by enforcing the conditioning as a requirement on these models.

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Section: Unique Challenges In Data-driven Reynolds Stress Closuresmentioning

confidence: 99%

Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

et al. 2019

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“…As universal function approximators, DNNs excel at settings where both the input and output are high-dimensional. Applications in flow simulations include pressure projections in solving Navier-Stokes equations [8], fluid flow through random heterogeneous media [9, 10,11], Reynolds-Averaged Navier-Stokes simulations [12,13,14] and others. Uncertainty quantification for DNNs is often studied under the re-emerging framework of Bayesian deep learning 1 [15], mostly using variational inference for approximate posterior of model parameters, e.g.…”

Section: Introductionmentioning

confidence: 99%

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Zhu

Zabaras

Koutsourelakis

et al. 2019

Journal of Computational Physics

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812

458

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Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.

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“…Bayesian inference has been used to optimize the coefficients of RANS models [61,62], as well as to derive functions that quantify and reduce the gap between the model predictions and high-fidelity data, such as DNS data [63][64][65]. Similarly, a Bayesian Neural Network (BNN) has been used to improve the predictions of RANS models and to specify the uncertainty associated with the prediction [66]. Random forests have also been trained on high-fidelity data to quantify the error of RANS models based on mean flow features [67,68].…”

Section: Simulationsmentioning

confidence: 99%

Machine-Learning Methods for Computational Science and Engineering

2020

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The re-kindled fascination in machine learning (ML), observed over the last few decades, has also percolated into natural sciences and engineering. ML algorithms are now used in scientific computing, as well as in data-mining and processing. In this paper, we provide a review of the state-of-the-art in ML for computational science and engineering. We discuss ways of using ML to speed up or improve the quality of simulation techniques such as computational fluid dynamics, molecular dynamics, and structural analysis. We explore the ability of ML to produce computationally efficient surrogate models of physical applications that circumvent the need for the more expensive simulation techniques entirely. We also discuss how ML can be used to process large amounts of data, using as examples many different scientific fields, such as engineering, medicine, astronomy and computing. Finally, we review how ML has been used to create more realistic and responsive virtual reality applications.

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Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

Cited by 110 publications

References 46 publications

Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Machine-Learning Methods for Computational Science and Engineering

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