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.

A novel approach for numerically propagating acoustic waves in two-dimensional quiescent media has been developed through a fully convolutional multi-scale neural network. This datadriven method managed to produce accurate results for long simulation times with a database of Lattice Boltzmann temporal simulations of propagating Gaussian Pulses, even in the case of initial conditions unseen during training time, such as the plane wave configuration or the two initial Gaussian pulses of opposed amplitudes. Two different choices of optimization objectives are compared, resulting in an improved prediction accuracy when adding the spatial gradient difference error to the traditional mean squared error loss function. Further accuracy gains are observed when performing an a posteriori correction on the neural network prediction based on the conservation of acoustic energy, indicating the benefit of including physical information in data-driven methods. Nomenclature= Gaussian pulse half-width 0

Accurate modeling of boundary conditions is crucial in computational physics. The ever increasing use of neural networks as surrogates for physics-related problems calls for an improved understanding of boundary condition treatment, and its influence on the network accuracy. In this paper, several strategies to impose boundary conditions (namely padding, improved spatial context, and explicit encoding of physical boundaries) are investigated in the context of fully convolutional networks applied to recurrent tasks. These strategies are evaluated on two spatio-temporal evolving problems modeled by partial differential equations: the 2D propagation of acoustic waves (hyperbolic PDE) and the heat equation (parabolic PDE). Results reveal a high sensitivity of both accuracy and stability on the boundary implementation in such recurrent tasks. It is then demonstrated that the choice of the optimal padding strategy is directly linked to the data semantics. Furthermore, the inclusion of additional input spatial context or explicit physics-based rules allows a better handling of boundaries in particular for large number of recurrences, resulting in more robust and stable neural networks, while facilitating the design and versatility of such types of networks. 1