In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model transient systems with non-linear dynamics at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.

Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulations to increase their predictive accuracy. While several data-driven turbulence models have been reported, the quantification of the uncertainties introduced has mostly been neglected. Uncertainty quantification for such data-driven models is essential since their predictive capability rapidly declines as they are tested for flow physics that deviate from that in the training data. In this work, we propose a novel data-driven framework that not only improves RANS predictions but also provides probabilistic bounds for fluid quantities such as velocity and pressure. The uncertainties capture both model form uncertainty as well as epistemic uncertainty induced by the limited training data. An invariant Bayesian deep neural network is used to predict the anisotropic tensor component of the Reynolds stress. This model is trained using Stein variational gradient decent algorithm. The computed uncertainty on the Reynolds stress is propagated to the quantities of interest by vanilla Monte Carlo simulation. Results are presented for two test cases that differ geometrically from the training flows at several different Reynolds numbers. The prediction enhancement of the data-driven model is discussed as well as the associated probabilistic bounds for flow properties of interest. Ultimately this framework allows for a quantitative measurement of model confidence and uncertainty quantification for flows in which no high-fidelity observations or prior knowledge is available.

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