Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks

The governing equations for fluid behavior are typically the Navier-Stokes equations, which are solved using discretization approaches like finite difference, finite volume, or finite element methods. However, such computational fluid dynamics (CFD) simulations can be computationally intensive, especially for turbulent flow and complex geometries, and changing the geometry requires recomputing the entire simulation. Hence, there is a need for a quick surrogate model for CFD simulations. Such surrogate models encompass a variety of approaches, including linear reduced order models [10, 26], such as reduced basis [39] and proper orthogonal decomposition [43] models, as well as neural network-based models [9], like convolutional neural networks (CNNs) [5, 8, 19, 28, 34] and neural operators [24, 33]. In present work, we focus on using neural networks as an approximation for CFD simulations. Instead of relying on a large dataset, we leverage the known governing equations of fluids to construct a physics-aware loss function and train our model to satisfy these equations discretely. This approach has recently become increasingly popular and was applied to dense neural networks (DNN) to solve partial differential equations (PDEs) with little training data [41] or without training data [52] as well as inverse problems with limited training data [18, 41]. More recently, this idea was also applied to convolutional neural networks (CNN) by using physics-aware loss functions to solve PDEs [2, 6, 11, 45, 49, 57], upscale and denoise solutions [12, 21], generally improve the predictive quality of a model [48, 56], or learn PDEs from data [31, 32]. For a comprehensive overview on scientific machine learning (SciML), we refer to [3, 55].