{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T03:21:40Z","timestamp":1774063300355,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,25]],"date-time":"2024-02-25T00:00:00Z","timestamp":1708819200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Russian Science Foundation","award":["21-71-30023"],"award-info":[{"award-number":["21-71-30023"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This study introduces an innovative approach leveraging physics-informed neural networks (PINNs) for the efficient computation of blood flows at the boundaries of a four-vessel junction formed by a Fontan procedure. The methodology incorporates a 3D mesh generation technique based on the parameterization of the junction\u2019s geometry, coupled with an advanced physically regularized neural network architecture. Synthetic datasets are generated through stationary 3D Navier\u2013Stokes simulations within immobile boundaries, offering a precise alternative to resource-intensive computations. A comparative analysis of standard grid sampling and Latin hypercube sampling data generation methods is conducted, resulting in datasets comprising 1.1\u00d7104 and 5\u00d7103 samples, respectively. The following two families of feed-forward neural networks (FFNNs) are then compared: the conventional \u201cblack-box\u201d approach using mean squared error (MSE) and a physically informed FFNN employing a physically regularized loss function (PRLF), incorporating mass conservation law. The study demonstrates that combining PRLF with Latin hypercube sampling enables the rapid minimization of relative error (RE) when using a smaller dataset, achieving a relative error value of 6% on the test set. This approach offers a viable alternative to resource-intensive simulations, showcasing potential applications in patient-specific 1D network models of hemodynamics.<\/jats:p>","DOI":"10.3390\/computation12030041","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T08:59:01Z","timestamp":1708937941000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Physically Informed Deep Learning Technique for Estimating Blood Flow Parameters in Four-Vessel Junction after the Fontan Procedure"],"prefix":"10.3390","volume":"12","author":[{"given":"Alexander","family":"Isaev","sequence":"first","affiliation":[{"name":"Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, 119991 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6412-9829","authenticated-orcid":false,"given":"Tatiana","family":"Dobroserdova","sequence":"additional","affiliation":[{"name":"Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, 119991 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4709-4513","authenticated-orcid":false,"given":"Alexander","family":"Danilov","sequence":"additional","affiliation":[{"name":"Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, 119991 Moscow, Russia"},{"name":"Moscow Institute of Physics and Technology, 141701 Dolgoprudny, Russia"},{"name":"Institute of Computer Sciences and Mathematical Modelling, Sechenov University, 119992 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3406-9623","authenticated-orcid":false,"given":"Sergey","family":"Simakov","sequence":"additional","affiliation":[{"name":"Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, 119991 Moscow, Russia"},{"name":"Moscow Institute of Physics and Technology, 141701 Dolgoprudny, Russia"},{"name":"Institute of Computer Sciences and Mathematical Modelling, Sechenov University, 119992 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,25]]},"reference":[{"key":"ref_1","first-page":"174","article-title":"Patient-Specific Surgical Planning, Where Do We Stand? The Example of the Fontan Procedure","volume":"44","author":"Kurtcuoglu","year":"2015","journal-title":"Ann. Biomed. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1016\/j.jtcvs.2017.11.068","article-title":"Virtual surgical planning, flow simulation, and 3-dimensional electrospinning of patient-specific grafts to optimize Fontan hemodynamics","volume":"155","author":"Siallagan","year":"2018","journal-title":"J. Thorac. Cardiovasc. Surg."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.jtcvs.2014.08.069","article-title":"Flow simulations and validation for the first cohort of patients undergoing the Y-graft Fontan procedure","volume":"149","author":"Yang","year":"2015","journal-title":"J. Thorac. Cardiovasc. Surg."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/jfm.2016.803","article-title":"Deep learning in fluid dynamics","volume":"814","author":"Kutz","year":"2017","journal-title":"J. Fluid Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"11545","DOI":"10.3934\/mbe.2023512","article-title":"Investigation on aortic hemodynamics based on physics-informed neural network","volume":"20","author":"Du","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2285","DOI":"10.1109\/TMI.2022.3161653","article-title":"Physics-Informed Neural Networks for Brain Hemodynamic Predictions Using Medical Imaging","volume":"41","author":"Sarabian","year":"2022","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1002\/nme.2579","article-title":"Gmsh: A 3-D Finite Element Mesh Generator with Built-in Pre- and Post-Processing Facilities","volume":"79","author":"Geuzaine","year":"2009","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_8","unstructured":"Kingma, D., and Adam, J.B. (2014, January 14\u201316). A Method for Stochastic Optimization. Proceedings of the 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada."},{"key":"ref_9","first-page":"1468","article-title":"An adaptive algorithm for quasioptimal mesh generation","volume":"39","author":"Vassilevski","year":"1999","journal-title":"Comp. Math. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1137\/S0895479899358194","article-title":"A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling","volume":"23","author":"Amestoy","year":"2001","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_11","unstructured":"Vassilevski, Y., Olshanskii, M., Simakov, S., Kolobov, A., and Danilov, A. (2020). Personalized Computational Hemodynamics. Models, Methods, and Applications for Vascular Surgery and Antitumor Therapy, Academic Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/S0951-8320(03)00058-9","article-title":"Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems","volume":"81","author":"Helton","year":"2003","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1109\/45.329294","article-title":"Feed-forward neural networks","volume":"13","author":"Bebis","year":"1994","journal-title":"IEEE Potentials"},{"key":"ref_14","first-page":"1929","article-title":"Dropout: A Simple Way to Prevent Neural Networks from Overfitting","volume":"15","author":"Srivastava","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/S0020-0255(96)00200-9","article-title":"The generalized sigmoid activation function: Competitive supervised learning","volume":"99","author":"Sridhar","year":"1997","journal-title":"Inf. Sci."},{"key":"ref_16","unstructured":"Nair, V., and Hinton, G. (2010, January 21\u201324). Rectified Linear Units Improve Restricted Boltzmann Machines. Proceedings of the International Conference on Machine Learning, Haifa, Israel."},{"key":"ref_17","unstructured":"Pretorius, A., Barnard, E., and Davel, M. (2019). Fundamentals of Artificial Intelligence Research, Springer."},{"key":"ref_18","unstructured":"Nwankpa, C., Ijomah, W., Gachagan, A., and Marshall, S. (2018). Activation Functions: Comparison of trends in Practice and Research for Deep Learning. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1145\/3065386","article-title":"ImageNet classification with deep convolutional neural networks","volume":"60","author":"Krizhevsky","year":"2012","journal-title":"Commun. ACM"},{"key":"ref_20","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning, MIT Press. Adaptive Computation and Machine Learning."},{"key":"ref_21","unstructured":"Amin, M., and Meidani, H. (2018). Physics-Informed Regularization of Deep Neural Networks. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Rojas, R. (1996). Neural Networks: A Systematic Introduction, Springer.","DOI":"10.1007\/978-3-642-61068-4"},{"key":"ref_23","unstructured":"Claesen, M., and De Moor, B. (2015). Hyperparameter Search in Machine Learning. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/0098-3004(93)90090-R","article-title":"Principal Components Analysis (PCA)","volume":"19","author":"Ratajczak","year":"1993","journal-title":"Comput. Geosci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1162391","DOI":"10.3389\/fphys.2023.1162391","article-title":"An anatomically detailed arterial-venous network model. Cerebral and coronary circulation","volume":"14","author":"Watanabe","year":"2023","journal-title":"Front. Physiol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1515\/rnam-2021-0025","article-title":"Numerical evaluation of the effectiveness of coronary revascularization","volume":"36","author":"Simakov","year":"2021","journal-title":"Russ. J. Numer. Anal. Math. Model."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5334\/gh.837","article-title":"Noninvasive Assessment of the Fractional Flow Reserve with the CT FFRc 1D Method: Final Results of a Pilot Study","volume":"16","author":"Gognieva","year":"2021","journal-title":"Glob. Heart"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:04:45Z","timestamp":1760105085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,25]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["computation12030041"],"URL":"https:\/\/doi.org\/10.3390\/computation12030041","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,25]]}}}