## 5 Conclusions  
This paper presents a novel approach to solve the Navier-Stokes equations using Physics-Informed Neural Networks. Three DNN paradigms (Fourier Neural Network, Convolutional LSTM and CNN-LSTM) were implemented and tested against the CFD solution of the cavity lid-driven flow problem. We used several Reynolds numbers to test and forecast the Re = 10,000 case, aiming to benchmark the Fourier Neural Network (FNO). The advantage of this new class of DNN architecture is the mapping of fields between different spaces (not necessarily Euclidean), which allows the learning of the parametric dependence of the solutions directly to an entire family of PDEs. From the results, the following conclusions can be highlighted:  
* A RANS k-\(\varepsilon\) CFD solution was performed to generate data (training and testing) to be fed to the models. A comparison with the results found in the literature was able to attest the data quality. The evaluation of the k-\(\varepsilon\) turbulence model against a full Direct Numerical Simulation indicated that the simpler CFD model, for this simple case, accurately represented the turbulence phenomena.
* After the tests for the models' architectures setup, the FNO and ConvLSTM paradigms performed better, with a consistent small advantage of FNO.
* With the selected models' architectures, a custom error parcel regarding the mass conservation error was added to the training step, using several weight values. Even though the RMSE of the test case did not improve, the resultant fields presented a notable improvement in physical coherence.
* The FNO paradigm was finally assessed to predict the solutions of the flow under several input/output situations, giving a testing RMSE of 0.008792 m/s for the best configuration (three timesteps for the input and three timesteps for the output), which was at least two orders of magnitude of the reference lid velocity (1.0 m/s).  
For the tested case, the FNO was able to consistently perform better than the other models, which qualifies it as an option to further develop DNN solutions of partial differential equations. Although presenting good performance, the FNO has some limitations, as presented by [PERSON] et al. [28], and DeepONet is an option to address such hindrances. For further work, implementation of the momentum conservation equation to the model could be performed to assess its contribution over the model's results, as well the effect of different image resolutions in the final model's prediction. Moreover, a deeper understanding of FNO limitations can be assessed by testing other problem configurations and the effect of turbulence levels, not only for RANS models, but also for Large-eddy Simulations or Direct Numerical Simulation (DNS) results. For instance, other flow geometries can be tested, which would produce more transient solutions of the PDEs, such as a bluff body or wake flows. Moreover, greater Reynolds numbers can be considered, which may be indicative of environmental flows.  
Conceptualization, J.V.G.T. and B.G.; methodology, P.A.C.R., J.V.G.T. and B.G.; software, P.A.C.R. and S.J.J.; validation, P.A.C.R., S.J.J., J.V.G.T. and B.G.; formal analysis, P.A.C.R. and S.J.J.; investigation, P.A.C.R., S.J.J., J.V.G.T. and B.G.; resources, J.V.G.T. and B.G.; data curation, J.V.G.T. and B.G.; writing--original draft preparation, P.A.C.R. and S.J.J. and A.A.A.; writing--review and editing, P.A.C.R., S.J.J., V.O.S., A.A.A., J.V.G.T. and B.G.; visualization, P.A.C.R., S.J.J., V.O.S. and A.A.A.; supervision, J.V.G.T. and B.G.; project administration, J.V.G.T. and B.G.; funding acquisition, B.G and J.V.G.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) Alliance, Grant No. 401643, in association with Lakes Environmental Software Inc., and by the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico--Brasil (CNPq), Grant No. 305456/2019-9.  
Not applicable.