Towards an hybrid computational strategy based on Deep Learning for incompressible flows

Abstract

The Poisson equation is present in very different domains of physics and engineering. In most cases, this equation can not be solved directly and iterative solvers are used. For many solvers, this step is computationally intensive. In this study, an alternative resolution method based on neural networks is evaluated for incompressible flows. A fluid solver coupled with a Convolutional Neural Network is developed and trained on random cases with constant density to predict the pressure field. Its performance is tested in a plume configuration, with different buoyancy forces, parametrized by the Richardson number. The neural network is compared to a traditional Jacobi solver. The performance improvement is considerable, although the accuracy of the network is found to depend on the flow operating point: low errors are obtained at low Richardson numbers, whereas the fluid solver becomes unstable with large errors for large Richardson number. Finally, a hybrid strategy is proposed in order to benefit from the calculation acceleration while ensuring a user-defined accuracy level. In particular, this hybrid CFD-NN strategy, by maintaining the desired accuracy whatever the flow condition, makes the code stable and reliable even at large Richardson numbers for which the network was not trained for. This study demonstrates the capability of the hybrid approach to tackle new flow physics, unseen during the network training.

Performance and accuracy assessments of an incompressible fluid solver coupled with a deep convolutional neural network

“…First, the four networks described in Section 3.1 are tested on a 2D plume test case with a Richardson number R i = 14:8, in order to evaluate the network accuracy and its generalization capabilities (since the training dataset was obtained for R i = 0). Following the previous work of Ajuria Illarramendi et al (2020), flows at high Richardson numbers can become numerically unstable and yield nonsymmetric simulations. Thus, R i = 14:8 is chosen, corresponding to such sensible regime, where the network prediction can become critical for a stable and reliable flow simulation.…”

“…However, slightly different plume developments are obtained depending on the network architecture. As introduced in the previous work of Ajuria Illarramendi et al (2020), the Jacobi solver is taken as the reference for the two studied test cases. The accuracy of the Jacobi solver solution increases with the number of iterations, but as the convergence rate does not follow a linear behavior, a trade-off between the number of iterations and the desired accuracy is usually necessary.…”

Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh

“…The domain was kept identical to the one in Section 8.2, but was discretised with 100 Â 100 grid points. Similar to the "hybrid" strategy applied by Ajuria Illarramendi et al (2020), designed to reduce the accumulation of errors during the time marching process, traditional solver iterations are applied to the output of the model. Table 4 displays the L 2 error norms of the velocities and the pressure compared to the analytical result at the end of the simulation (t ¼ 1:0).…”

“…Moreover, the former approach trains the NN to minimize the divergence of the velocity field only while the latter adopts a more direct strategy by trying to instead minimize a linear combination of the L2 norms of the velocity divergence and the discrepancy between the predicted and ground truth pressure correction values by leveraging the additional data available from the specific methodology. Building on the strategy developed in these works, Ajuria Illarramendi et al (2020) used a CNN to handle the Poisson solver step of CFD simulations of plumes and flows around cylinders, demonstrating stable and accurate time evolution even for Richardson numbers greater than in the training data when applied in combination with several Jacobi iterations. Outside fluid mechanics, Shan et al (2017) investigated the application of a fully convolutional NN to predict the electric potential on cubic 64 Â 64 Â 64 grids given the charge distributions and (constant) permittivities, claiming average relative errors below 3% and speedups compared to traditional methods.…”

Generalization Capability of Convolutional Neural Networks for Progress Variable Variance and Reaction Rate Subgrid-Scale Modeling

“…Skip connections also allow the network to combine low-level features learned by the shallow layers of the encoder with the more complex, abstract features learned by the decoder, and accelerate training convergence [56]. Multiscale CNN architectures for field-to-field fluid predictions were successfully used by Lapeyre et al [4] to model SGS flame wrinkling and Ajuria et al [58] to solve the Poisson equation in incompressible flows. In the following, the U-Net model will simply be called the CNN.…”