{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T22:13:04Z","timestamp":1790979184392,"version":"4.1.0"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Key Program Special Fund in XJTLU","award":["KSF-E-50"],"award-info":[{"award-number":["KSF-E-50"]}]},{"name":"the Key Program Special Fund in XJTLU","award":["PGRS1912009"],"award-info":[{"award-number":["PGRS1912009"]}]},{"name":"the Key Program Special Fund in XJTLU","award":["RDF-19-01-15"],"award-info":[{"award-number":["RDF-19-01-15"]}]},{"name":"XJTLU Postgraduate Research Scholarship","award":["KSF-E-50"],"award-info":[{"award-number":["KSF-E-50"]}]},{"name":"XJTLU Postgraduate Research Scholarship","award":["PGRS1912009"],"award-info":[{"award-number":["PGRS1912009"]}]},{"name":"XJTLU Postgraduate Research Scholarship","award":["RDF-19-01-15"],"award-info":[{"award-number":["RDF-19-01-15"]}]},{"name":"XJTLU Research Development Funding","award":["KSF-E-50"],"award-info":[{"award-number":["KSF-E-50"]}]},{"name":"XJTLU Research Development Funding","award":["PGRS1912009"],"award-info":[{"award-number":["PGRS1912009"]}]},{"name":"XJTLU Research Development Funding","award":["RDF-19-01-15"],"award-info":[{"award-number":["RDF-19-01-15"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Machine Learning (ML) and\/or Deep Learning (DL) methods can be used to predict fluid flow in porous media, as a suitable replacement for classical numerical approaches. Such data-driven approaches attempt to learn mappings between finite-dimensional Euclidean spaces. A novel neural framework, named Fourier Neural Operator (FNO), has been recently developed to act on infinite-dimensional spaces. A high proportion of the research available on the FNO has focused on problems with large-shape data. Furthermore, most published studies apply the FNO method to existing datasets. This paper applies and evaluates FNO to predict pressure distribution over a small, specified shape-data problem using 1700 Finite Element Method (FEM) generated samples, from heterogeneous permeability fields as the input. Considering FEM-calculated outputs as the true values, the configured FNO model provides superior prediction performance to that of a Convolutional Neural Network (CNN) in terms of statistical error assessment based on the coefficient of determination (R2) and Mean Squared Error (MSE). Sensitivity analysis considering a range of FNO configurations reveals that the most accurate model is obtained using modes=15 and width=100. Graphically, the FNO model precisely follows the observed trend in each porous medium evaluated. There is potential to further improve the FNO\u2019s performance by including physics constraints in its network configuration.<\/jats:p>","DOI":"10.3390\/a16010024","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T03:00:59Z","timestamp":1672628459000},"page":"24","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Fourier Neural Operator for Fluid Flow in Small-Shape 2D Simulated Porous Media Dataset"],"prefix":"10.3390","volume":"16","author":[{"given":"Abouzar","family":"Choubineh","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Liverpool, Liverpool L69 7ZX, UK"},{"name":"Department of Applied Mathematics, Xi\u2019an Jiaotong-Liverpool University, Suzhou 215123, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1036-007X","authenticated-orcid":false,"given":"Jie","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an Jiaotong-Liverpool University, Suzhou 215123, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3202-4069","authenticated-orcid":false,"given":"David A.","family":"Wood","sequence":"additional","affiliation":[{"name":"DWA Energy Limited, Lincoln LN5 9JP, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frans","family":"Coenen","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Liverpool, Liverpool L69 7ZX, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fei","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Xi\u2019an Jiaotong-Liverpool University, Suzhou 215123, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,1]]},"reference":[{"key":"ref_1","unstructured":"Golub, G.H., and Ortega, J.M. (1992). Scientific Computing and Differential Equations: An Introduction to Numerical Methods, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1007\/s40996-021-00698-5","article-title":"Finite difference modelings of groundwater flow for constructing artificial recharge structures","volume":"46","author":"Tao","year":"2022","journal-title":"Iran. J. Sci. Technol. Trans. Civ. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"100187","DOI":"10.1016\/j.ijft.2022.100187","article-title":"Applying numerical and computational methods to investigate the changes in the fluid parameters of the fluid passing over fins of different shapes with the finite element method","volume":"15","author":"Fathollahi","year":"2022","journal-title":"Int. J. Thermofluids"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1891","DOI":"10.1007\/s10973-021-10637-1","article-title":"Parallel finite volume method-based fluid flow computations using OpenMP and CUDA applying different schemes","volume":"145","author":"Afzal","year":"2021","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2150246","DOI":"10.1142\/S0218348X21502467","article-title":"Numerical Solutions of Space Fractional Variable-Coefficient Kdv\u2013Modified Kdv Equation by Fourier Spectral Method","volume":"29","author":"Han","year":"2021","journal-title":"Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.apnum.2020.09.019","article-title":"A meshless method for time fractional nonlinear mixed diffusion and diffusion-wave equation","volume":"160","author":"Bhardwaj","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1080\/10916466.2018.1447960","article-title":"Application of ANFIS-GA algorithm for forecasting oil flocculated asphaltene weight percentage in different operation conditions","volume":"36","author":"Keybondorian","year":"2018","journal-title":"Pet. Sci. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1016\/j.petrol.2019.03.003","article-title":"Asphaltene adsorption using green nanocomposites: Experimental study and adaptive neuro-fuzzy interference system modeling","volume":"177","author":"Mohammadi","year":"2019","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"13478","DOI":"10.1021\/acs.chemrev.2c00061","article-title":"Machine learning for electrocatalyst and photocatalyst design and discovery","volume":"122","author":"Mai","year":"2022","journal-title":"Chem. Rev."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kazemi, P., Ghisi, A., and Mariani, S. (2022). Classification of the Structural Behavior of Tall Buildings with a Diagrid Structure: A Machine Learning-Based Approach. Algorithms, 15.","DOI":"10.3390\/a15100349"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, W., Wang, S., Zhang, X., Yao, L., Yue, L., Qian, B., and Li, X. (2018, January 3\u20135). EEG-based motion intention recognition via multi-task RNNs. Proceedings of the 2018 SIAM International Conference on Data Mining, SIAM, San Diego, CA, USA.","DOI":"10.1137\/1.9781611975321.32"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110751","DOI":"10.1016\/j.petrol.2022.110751","article-title":"An innovative application of deep learning in multiscale modeling of subsurface fluid flow: Reconstructing the basis functions of the mixed GMsFEM","volume":"216","author":"Choubineh","year":"2022","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"014502","DOI":"10.1115\/1.4054868","article-title":"A quantitative insight into the role of skip connections in deep neural networks of low complexity: A case study directed at fluid flow modeling","volume":"23","author":"Choubineh","year":"2022","journal-title":"J. Comput. Inf. Sci. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Pawar, P., Ainapure, B., Rashid, M., Ahmad, N., Alotaibi, A., and Alshamrani, S.S. (2022). Deep Learning Approach for the Detection of Noise Type in Ancient Images. Sustainability, 14.","DOI":"10.3390\/su141811786"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mijalkovic, J., and Spognardi, A. (2022). Reducing the False Negative Rate in Deep Learning Based Network Intrusion Detection Systems. Algorithms, 15.","DOI":"10.3390\/a15080258"},{"key":"ref_16","unstructured":"Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. (2020). Fourier neural operator for parametric partial differential equations. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Gallant, A.R., and White, H. (1988, January 24\u201327). There exists a neural network that does not make avoidable mistakes. Proceedings of the ICNN, San Diego, CA, USA.","DOI":"10.1109\/ICNN.1988.23903"},{"key":"ref_18","unstructured":"Silvescu, A. (1999, January 10\u201316). Fourier neural networks. Proceedings of the IJCNN\u201999, International Joint Conference on Neural Networks, Proceedings (Cat. No. 99CH36339), Washington, DC, USA."},{"key":"ref_19","unstructured":"Liu, S. (2013, January 14\u201317). Fourier neural network for machine learning. Proceedings of the 2013 International Conference on Machine Learning and Cybernetics, Tianjin, China."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104180","DOI":"10.1016\/j.advwatres.2022.104180","article-title":"U-FNO\u2014An enhanced Fourier neural operator-based deep-learning model for multiphase flow","volume":"163","author":"Wen","year":"2022","journal-title":"Adv. Water Resour."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Rashid, M.M., Pittie, T., Chakraborty, S., and Krishnan, N.A. (2022). Learning the stress-strain fields in digital composites using fourier neural operator. iScience, 105452.","DOI":"10.1016\/j.isci.2022.105452"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, Z., Peng, W., Yuan, Z., and Wang, J. (2022). Fourier neural operator approach to large eddy simulation of three-dimensional turbulence. Theor. Appl. Mech. Lett., 100389.","DOI":"10.1016\/j.taml.2022.100389"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Johnny, W., Brigido, H., Ladeira, M., and Souza, J.C.F. (2022, January 22\u201325). Fourier Neural Operator for Image Classification. Proceedings of the 2022 17th Iberian Conference on Information Systems and Technologies (CISTI), Madrid, Spain.","DOI":"10.23919\/CISTI54924.2022.9820128"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Chen, Z. (2007). Reservoir Simulation: Mathematical Techniques in Oil Recovery, SIAM.","DOI":"10.1137\/1.9780898717075"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1109\/T-C.1970.222918","article-title":"Application of the Karhunen-Loeve expansion to feature selection and ordering","volume":"100","author":"Fukunaga","year":"1970","journal-title":"IEEE Trans. Comput."},{"key":"ref_26","unstructured":"Lasser, R. (1996). Introduction to Fourier Series, CRC Press."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Strichartz, R.S. (2003). A Guide to Distribution Theory and Fourier Transforms, World Scientific Publishing Company.","DOI":"10.1142\/5314"},{"key":"ref_28","unstructured":"Subramanian, V. (2018). Deep Learning with PyTorch: A Practical Approach to Building Neural Network Models Using PyTorch, Packt Publishing Ltd."},{"key":"ref_29","unstructured":"Reddi, S.J., Kale, S., and Kumar, S. (2019). On the convergence of adam and beyond. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Nussbaumer, H.J. (1981). The fast Fourier transform. Fast Fourier Transform and Convolution Algorithms, Springer.","DOI":"10.1007\/978-3-662-00551-4"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Joseph, F.J.J., Nonsiri, S., and Monsakul, A. (2021). Keras and TensorFlow: A hands-on experience. Advanced Deep Learning for Engineers and Scientists, Springer.","DOI":"10.1007\/978-3-030-66519-7_4"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/1\/24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:55:32Z","timestamp":1760118932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/1\/24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["a16010024"],"URL":"https:\/\/doi.org\/10.3390\/a16010024","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,1]]}}}