Numerical solution of partial differential equations (PDEs) has been a very active field of research in the last recent, when in particular Finite Element Method (FEM) was primarily studied. FEM involves discretizing a continuous function space using a discrete mesh and solving high-dimensional a linear system, which can be computationally demanding Dhatt et al. [2012]. This computational cost becomes particularly substantial in tasks requiring repetitive solutions. An example of such tasks is engineering design which necessitates solving parameterized PDEs over a wide range of PDE parameters and domain geometries for design evaluation Almasri et al. [2024]. Recently, machine learning techniques have been introduced to accelerate the process of solving PDEs by learning a neural operator as a mapping from variable PDE parameters and/or domain geometry to the PDE solution Lu et al. [2021]. Once a neural operator model is successfully trained on a dataset, it can generalize to new, unseen parameters and domain geometries. This is done by a single forward pass through the trained neural network, with minimal computational cost.

Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer aided aerodynamic design of automobiles or airplanes to simulations of physical effects in CGI to research in meteorology. Recent differentiable fluid simulations allow gradient based methods to optimize e.g. fluid control systems in an informed manner. Solving the partial differential equations governed by the dynamics of the underlying physical systems, however, is a challenging task and current numerical approximation schemes still come at high computational costs. In this work, we propose an unsupervised framework that allows powerful deep neural networks to learn the dynamics of incompressible fluids end to end on a grid-based representation. For this purpose, we introduce a loss function that penalizes residuals of the incompressible Navier Stokes equations. After training, the framework yields models that are capable of fast and differentiable fluid simulations and can handle various fluid phenomena such as the Magnus effect and K\'arm\'an vortex streets. Besides demonstrating its real-time capability on a GPU, we exploit our approach in a control optimization scenario.