## 1 Introduction  
Throughout history, numerical modeling has emerged as an important tool for technological development; it is used for nearly all human innovations. In the ecosystem of numerical models, Computational Fluid Dynamics (CFD) is a class of tools that have dominated many scientific and engineering fields, from microfluid flows [1; 2] to large scale flows [3; 4; 5], as well as specific device evaluation [6]. Despite this success, modern CFD approaches have drawbacks that limit their effectiveness. For example, there is no unique and general-purpose turbulence closure model, and the computational cost, in terms of time and CPU requirements, is usually very high, even for simple problems [7; 8]. In this context, Machine Learning (ML) models appear as a candidate to overcome some of those issues. Recent works have shown that the ML paradigm can be employed in different problems, such as microscale transport between fluids [9], fluid flow through nanochannels [10], enzyme production optimization [11] and solid fuel performance prediction [12], attesting its multidisciplinary applications. Recently, Deep Neural Networks (DNNs) have been used to solve Partial Differential Equations (PDEs) of related problems [13; 14; 15]. In this context, CFD studies have emerged as a natural path for implementing and testing those techniques. The examples are many, and they come because the evaluation of DNN models are applied to directly map the PDEsolution [16; 17; 18; 19], as well as the insertion of ML models in parts of the PDE solution [15; 20], especially for the closure problem [13; 21]. The research reports solution acceleration in the order of 40-80 times [22]. As a recent advance in the use of DNNs to solve PDE-related problems, a new approach has been developed by researchers in the field. It consists of embedding physical information into the DNN model, which can be achieved in several ways [23], improving the model's computational performance when compared with traditional finite element methods [24]. One of these new approaches includes the insertion of physical constraints directly in the DNN models, which is known as a hard constraint. One can also impose a soft constraint, whereby the loss function used to assess the model performance is augmented to impose physical restraints [25]. Both practices were reported to improve accuracy and generalizability, while also reducing training times [26]. In this work, we applied both approaches, where the soft constraints were imposed by the boundary conditions (BCs), as well as the mass conservation law to assess its contribution to the model's predictions. On the other hand, as a hard constraint, we tested the Fourier Neural Operator (FNO) [27], which is a type of Neural Network (NN) layer with high physical appeal. It pertains to a new class of NN layers, which maps fields between different spaces (not necessarily Euclidean) and thus can learn the parametric dependence of the solution directly to an entire family of PDEs. To our best knowledge, there are still few results in the literature regarding the implementation and testing of FNO for turbulent flow simulations [28]. It is common in CFD studies to test and assess the proposed methodology on classical problems [29; 30], whose main characteristics are their ubiquitous validation by the community, their simplicity to be set, their representation of complex characteristics and their indication of real-world problems. Considering this, we chose the lid-driven cavity flow problem to be modeled by the proposed DNNs. The main objective of this work was to calibrate and test different DNN paradigms for the simulation of a flow under laminar and turbulent regimes. We tested their behavior by adding soft (boundary conditions and mass conservation law) and hard (Fourier Neural Operator) constraints. Finally, we discovered and assessed the best model, its hyperparameters and setup, including the number of timesteps to use when learning the flow dynamics.