Deep Neural Network Modeling for CFD Simulations: Benchmarking the Fourier Neural Operator on the Lid-Driven Cavity Case

Round 1

Reviewer 1 Report

This is a well-written paper that has done numerical experiments using mainly the ConvLSTM model and the Fourier Neural Operator.  I have a a few suggestions on how to improve the paper but in general, I thought this paper was acceptable.

 

I think the authors could do a better job of defining the goals of the model.  Predicting one time step given the three previous time steps is not that difficult (depending on the time step).  For example, a constant extrapolation only incurs errors that are first order with respect to the time step.  What makes doing all of this work useful?

 

Secondly, the authors didn't do a good job demonstrating the unsteady nature of the flow.  It was hard to tell from the figures how the flow changed in time.  Does it reach a completely steady state?  In that case, predicting with a constant extrapolation would be perfect.  The authors need to provide some information on the unsteady nature of the flow.

 

Third, there was a lot of Machine Learning (ML) jargon in the paper that wasn't explained.  I had to look up the definitions of LSTM online for example.  The authors should go through the paper and make sure that at least all acronyms are defined.  It would be nice if a brief explanation of some of the ML terminology was given too (what is an epoch for example)

 

To me, it is odd to try to add the mass conservation constraint without talking about momentum conservation.  I don't see why you would apply one and not the other.  The authors should at least discuss this point somewhere.

 

I noticed in Figure 7, that the ground truth does not satisfy mass conservation.  The numerical simulations should satisfy this requirement.  Maybe the way the constraint is implemented is incompatible with the way it is enforced in the numerical code.  Can this be fixed?

 

The root mean square errors and the mass conservation errors in particular have many leading zeros.  It would be easier to compare the number if they were written in scientific notation.

 

Author Response

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Reviewer 2 Report

The authors have presented the article in well manner, I recommend it for publication.

Author Response

Dear reviewer,

Thank you for your time reviewing our manuscript. We are glad to know that we could transmit our work in a clear way for your understanding.

Best regards,

The Authors. 

Reviewer 3 Report

Comments on manuscript No: applsci-2213542

The authors thoroughly compare ConvLSTM, CNN-LSTM, and Fourier Neural Operator (FNO) in solving partial differential equations. The results are impressive and demonstrate the potential of the proposed FNO approach. The paper provides a clear and detailed description of the methodology and the results obtained. The comparison between the three methods is well presented and provides valuable insights. The proposed FNO method is novel and innovative, offering a promising alternative to existing approaches for solving partial differential equations. The results of the experiments show that FNO outperforms both ConvLSTM and CNN-LSTM in terms of accuracy and efficiency. Overall, this is a well-written and well-structured paper that makes a valuable contribution to the field of machine learning for partial differential equation solving. The proposed FNO method shows great promise and is worth further investigation. The authors should consider expanding their analysis to multi-step predictions to further demonstrate the effectiveness of the FNO model.

1.      Line 119, ‘DL models were built and tested’. Which package/application was used to create these models?

2.      The paper could be strengthened by providing a more in-depth analysis of the computational time and stability of the proposed FNO method. A discussion of the limitations and potential future work would also be useful.

3.      It would be interesting to see a deeper analysis of the impact of different hyperparameters on the performance of ConvLSTM, and how it affects the overall accuracy of the turbulence model predictions.

4.      The use of OpenFOAM 2D images highlights the potential of using simulation data as a source for training machine learning models. The authors should consider exploring the impact of different image resolutions or feature extraction techniques on the performance of the model.

5.      The limitations of the proposed FNO model should be discussed in greater detail, and future work should aim to address these limitations.

6.      Introduction part should be improved with more number of pertinent papers published recently on similar research paradigm. For example, the following papers could enrich the quality of the literature review.

·         Prediction of Electrodiffusioosmotic Transport of Shear-thinning Fluids in a Nanochannel using Artificial Neural Network, Physics of Fluids, Volume 35, pp. 012018, January 2023, DOI: 10.1063/5.0134432

·         Application of Artificial Neural Network for Understanding Multi-Layer Microscale Transport Comprising of Alternate Newtonian and non-Newtonian fluids, Colloids Surfaces A.  Physicochem. Eng. Asp., vol 642, pp. 128664, 5 June 2022, doi: 10.1016/j.colsurfa.2022

·         Artificial neural network-based modelling of optimized experimental study of xylanase production by Penicillium citrinum xym2." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering vol. 236, no. 4, 1340-1348, December 2021, doi: 10.1177/09544089211064153

G. Kumar, Gireeshkumaran Thampi B. S, P. K. Mondal, Predicting Performance of Briquette Made from Millet Bran: A Neural Network 

Author Response

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Reviewer 4 Report

The authors present a deep neural network modelling for CFD simulations.  The paper provides a useful insight to utilise FNO, Conv-LSTM and CNN-LSTM for solving the partial differential equations of the RANS turbulence model for the 2D lid-driven cavity flow. This is a well-structured paper however; the authors need to add enough information to the paper particularly for Conv-LSTM and FNO methods. 

Some other comments are as follows:

1- Please add governing equations for your numerical example, 2D lid-driven. What are the state variables (e.g., Velocities, Pressure).

-    2- It would be more reasonable to carry out couples of convergence studies. For example, showing RMSE vs number of collocation points for different methods (CNN-LSTM, Conv-LSTM, FNO).

-      3-FNO has couple of drawbacks [24] and It would be good to address them and mention to other alternatives such as DeepONet.   

 

 

4-Please correct the legends in figure 2 & 3. (should be RANS k-eplison not k-e)

[24]: Lu, Lu, et al. "A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data." Computer Methods in Applied Mechanics and Engineering 393 (2022): 114778.

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 3 Report

The authors have considered my comments in the revised version.