Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations

Abstract

We propose to accelerate a high order discontinuous Galerkin solver using neural networks. We include a corrective forcing to a low polynomial order simulation to enhance its accuracy. The forcing is obtained by training a deep fully connected neural network, using a high polynomial order simulation but only for a short time frame. With this corrective forcing, we can run the low polynomial order simulation faster (with large time steps and low cost per time step) while improving its accuracy. We explored this idea for a 1D Burgers' equation in (Marique and Ferrer, CAF 2022), and we have extended this work to the 3D Navier-Stokes equations, with and without a Large Eddy Simulation closure model. We test the methodology with the turbulent Taylor Green Vortex case and for various Reynolds numbers (30, 200 and 1600). In addition, the Taylor Green Vortex evolves with time and covers laminar, transitional, and turbulent regimes, as time progresses. The proposed methodology proves to be applicable to a variety of flows and regimes. The results show that the corrective forcing is effective in all Reynolds numbers and time frames (excluding the initial flow development). We can train the corrective forcing with a polynomial order of 8, to increase the accuracy of simulations from a polynomial order 3 to 6, when correcting outside the training time frame. The low order correct solution is 4 to 5 times faster than a simulation with comparable accuracy (polynomial order 6). Additionally, we explore changes in the hyperparameters and use transfer learning to speed up the training. We observe that it is not useful to train a corrective forcing using a different flow condition. However, an already trained corrective forcing can be used to initialise a new training (at the correct flow conditions) to obtain an effective forcing with only a few training iterations.

References in corpus (10)

• Machine learning accelerated computational fluid dynamics
• A Review of Deep Transfer Learning and Recent Advancements
• EXPANSE: A Deep Continual / Progressive Learning System for Deep Transfer Learning
• Enhancing Computational Fluid Dynamics with Machine Learning
• Improving aircraft performance using machine learning: a review
• From coarse wall measurements to turbulent velocity fields through deep learning
• A transformer-based synthetic-inflow generator for spatially-developing turbulent boundary layers
• HORSES3D: a high-order discontinuous Galerkin solver for flow simulations and multi-physics applications
• Learned Coarse Models for Efficient Turbulence Simulation
• Predicting the wall-shear stress and wall pressure through convolutional neural networks

Cited by in corpus (4)

• Improving aircraft performance using machine learning: a review
• Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations
• DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations
• A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries