摘要 :
The simulation of rarefied and/or nonequilibrium flows generally requires the computation of expensive and approximate solutions to the Boltzmann equations. Since these regimes are at the limit of the continuum assumption, the acc...
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The simulation of rarefied and/or nonequilibrium flows generally requires the computation of expensive and approximate solutions to the Boltzmann equations. Since these regimes are at the limit of the continuum assumption, the accuracy of the (computationally tractable) Navier-Stokes equations is unreliable. To be able to reliably use the Navier-Stokes equations in these regimes, a deep learning augmentation framework was recently presented [1] with optimization targets obtained from Boltzmann equation solutions. This framework is different from standard applications of deep learning methods to physical systems. It leverages the a-priori-known physics and ensures consistency with the governing equations. The training of the deep neural network is done via solving adjoint equations which calculate the the loss sensitivities with respect to the dependent variables. The adjoint PDEs are constructed using automatic differentiation (AD). To enable more powerful models, we propose entropy constraints on the closure model outputs to ensure consistency with the second law of thermodynamics. An example prediction of density profiles for low-pressure argon is presented, for which the augmented models have comparable accuracy to reference direct Boltzmann solutions. Comparisons of different augmentation methods are also presented.
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摘要 :
The predictive accuracy of the Navier-Stokes equations is known to degrade at the limits of the continuum assumption, for example, in rarefied and/or nonequilibrium gases, thereby necessitating expensive and often highly approxima...
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The predictive accuracy of the Navier-Stokes equations is known to degrade at the limits of the continuum assumption, for example, in rarefied and/or nonequilibrium gases, thereby necessitating expensive and often highly approximate solutions to the Boltzmann equation. While tractable in one spatial dimension, their high dimensionality (n physical plus three phase-space) makes multi-dimensional Boltzmann calculations impractical for all but canonical configurations. It is therefore desirable to augment the accuracy of the Navier-Stokes equations in these regimes. We present an application of a deep learning (DL) method to extend the validity of the Navier-Stokes equations to the transitional flow regime. It works by encoding the "missing" (i.e., sub-continuum) physics in the Navier-Stokes equations via a neural network, which is trained by targeting density, velocity, and energy profiles obtained from direct Boltzmann solutions. While standard DL methods (e.g., those developed for image recognition, language processing, etc.) can be considered ad-hoc due to the absence of underlying physical laws, at least in the sense that the systems are not governed by known partial differential equations (PDEs), the DL framework applied here leverages the a priori-known Boltzmann physics while ensuring that the trained model is consistent with the Navier-Stokes PDEs. The online training procedure solves adjoint PDEs, which efficiently provide the gradient of the loss function with respect to the forward PDE solution. The adjoint PDEs are automatically constructed using algorithmic differentiation (AD). The model is trained and applied to predict shock thickness in low-pressure argon. The resulting DL-augmented Navier-Stokes equations have comparable accuracy to the target Boltzmann solutions. Extensions to other regimes and gas flows are discussed for future work.
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