DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)

DPM: A deep learning PDE augmentation method (with application to large-eddy simulation) Jonathan B. Freund, Jonathan F. MacArt †, and Justin Sirignano ‡§ November 22, 2019 Abstract Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is embedded in a partial differential equation (PDE) that expresses the known physics and learns to describe the corresponding unknown or unrepresented physics from the data. Crafted as such, the neural network can also provide corrections for erroneously represented physics, such as discretization errors associated with the PDE's numerical solution. Once trained, the deep learning PDE model (DPM) can make out-of-sample predictions for new physical parameters, geometries, and boundary conditions. Estimating the embedded neural network requires optimizing over the entire PDE, which itself is a function of the neural network. Adjoint partial differential equations are used to efficiently calculate the high-dimensional gradient of the objective function with respect to the neural network parameters. A stochastic adjoint method (SAM), similar in spirit to stochastic gradient descent, further accelerates training. The approach is demonstrated and evaluated for turbulence predictions using large-eddy simulation (LES), a filtered version of the Navier-Stokes equation containing unclosed sub-filter-scale terms. High-fidelity direct numerical simulations (DNS) of decaying isotropic turbulence provide the training and testing data. The DPM outperforms the widely-used constant-coefficient and dynamic Smagorinsky models, even for filter sizes so large that these established models become qualitatively incorrect. It also significantly outperforms a priori trained models, which do not account for the full PDE. For comparable accuracy, the overall cost is reduced. Simulations of the DPM are accelerated by efficient GPU implementations of network evaluations. Measures of discretization errors, which are well-known to be consequential in LES, suggest that the ability of the training formulation to correct for these errors Mechanical Science & Engineering and Aerospace Engineering, University of Illinois at Urbana-Champaign, jbfre-und@illinois.edu