{"title":"基于物理信息的神经网络解在主动脉瓣血流建模中的准确性和有效性评估","authors":"Jacques Francois Du Toit, R. Laubscher","doi":"10.3390/mca28020062","DOIUrl":null,"url":null,"abstract":"Physics-Informed Neural Networks (PINNs) are a new class of machine learning algorithms that are capable of accurately solving complex partial differential equations (PDEs) without training data. By introducing a new methodology for fluid simulation, PINNs provide the opportunity to address challenges that were previously intractable, such as PDE problems that are ill-posed. PINNs can also solve parameterized problems in a parallel manner, which results in favorable scaling of the associated computational cost. The full potential of the application of PINNs to solving fluid dynamics problems is still unknown, as the method is still in early development: many issues remain to be addressed, such as the numerical stiffness of the training dynamics, the shortage of methods for simulating turbulent flows and the uncertainty surrounding what model hyperparameters perform best. In this paper, we investigated the accuracy and efficiency of PINNs for modeling aortic transvalvular blood flow in the laminar and turbulent regimes, using various techniques from the literature to improve the simulation accuracy of PINNs. Almost no work has been published, to date, on solving turbulent flows using PINNs without training data, as this regime has proved difficult. This paper aims to address this gap in the literature, by providing an illustrative example of such an application. The simulation results are discussed, and compared to results from the Finite Volume Method (FVM). It is shown that PINNs can closely match the FVM solution for laminar flow, with normalized maximum velocity and normalized maximum pressure errors as low as 5.74% and 9.29%, respectively. The simulation of turbulent flow is shown to be a greater challenge, with normalized maximum velocity and normalized maximum pressure errors only as low as 41.8% and 113%, respectively.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Evaluation of Physics-Informed Neural Network Solution Accuracy and Efficiency for Modeling Aortic Transvalvular Blood Flow\",\"authors\":\"Jacques Francois Du Toit, R. Laubscher\",\"doi\":\"10.3390/mca28020062\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Physics-Informed Neural Networks (PINNs) are a new class of machine learning algorithms that are capable of accurately solving complex partial differential equations (PDEs) without training data. By introducing a new methodology for fluid simulation, PINNs provide the opportunity to address challenges that were previously intractable, such as PDE problems that are ill-posed. PINNs can also solve parameterized problems in a parallel manner, which results in favorable scaling of the associated computational cost. The full potential of the application of PINNs to solving fluid dynamics problems is still unknown, as the method is still in early development: many issues remain to be addressed, such as the numerical stiffness of the training dynamics, the shortage of methods for simulating turbulent flows and the uncertainty surrounding what model hyperparameters perform best. In this paper, we investigated the accuracy and efficiency of PINNs for modeling aortic transvalvular blood flow in the laminar and turbulent regimes, using various techniques from the literature to improve the simulation accuracy of PINNs. Almost no work has been published, to date, on solving turbulent flows using PINNs without training data, as this regime has proved difficult. This paper aims to address this gap in the literature, by providing an illustrative example of such an application. The simulation results are discussed, and compared to results from the Finite Volume Method (FVM). It is shown that PINNs can closely match the FVM solution for laminar flow, with normalized maximum velocity and normalized maximum pressure errors as low as 5.74% and 9.29%, respectively. The simulation of turbulent flow is shown to be a greater challenge, with normalized maximum velocity and normalized maximum pressure errors only as low as 41.8% and 113%, respectively.\",\"PeriodicalId\":53224,\"journal\":{\"name\":\"Mathematical & Computational Applications\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2023-04-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical & Computational Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/mca28020062\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical & Computational Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/mca28020062","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Evaluation of Physics-Informed Neural Network Solution Accuracy and Efficiency for Modeling Aortic Transvalvular Blood Flow
Physics-Informed Neural Networks (PINNs) are a new class of machine learning algorithms that are capable of accurately solving complex partial differential equations (PDEs) without training data. By introducing a new methodology for fluid simulation, PINNs provide the opportunity to address challenges that were previously intractable, such as PDE problems that are ill-posed. PINNs can also solve parameterized problems in a parallel manner, which results in favorable scaling of the associated computational cost. The full potential of the application of PINNs to solving fluid dynamics problems is still unknown, as the method is still in early development: many issues remain to be addressed, such as the numerical stiffness of the training dynamics, the shortage of methods for simulating turbulent flows and the uncertainty surrounding what model hyperparameters perform best. In this paper, we investigated the accuracy and efficiency of PINNs for modeling aortic transvalvular blood flow in the laminar and turbulent regimes, using various techniques from the literature to improve the simulation accuracy of PINNs. Almost no work has been published, to date, on solving turbulent flows using PINNs without training data, as this regime has proved difficult. This paper aims to address this gap in the literature, by providing an illustrative example of such an application. The simulation results are discussed, and compared to results from the Finite Volume Method (FVM). It is shown that PINNs can closely match the FVM solution for laminar flow, with normalized maximum velocity and normalized maximum pressure errors as low as 5.74% and 9.29%, respectively. The simulation of turbulent flow is shown to be a greater challenge, with normalized maximum velocity and normalized maximum pressure errors only as low as 41.8% and 113%, respectively.
期刊介绍:
Mathematical and Computational Applications (MCA) is devoted to original research in the field of engineering, natural sciences or social sciences where mathematical and/or computational techniques are necessary for solving specific problems. The aim of the journal is to provide a medium by which a wide range of experience can be exchanged among researchers from diverse fields such as engineering (electrical, mechanical, civil, industrial, aeronautical, nuclear etc.), natural sciences (physics, mathematics, chemistry, biology etc.) or social sciences (administrative sciences, economics, political sciences etc.). The papers may be theoretical where mathematics is used in a nontrivial way or computational or combination of both. Each paper submitted will be reviewed and only papers of highest quality that contain original ideas and research will be published. Papers containing only experimental techniques and abstract mathematics without any sign of application are discouraged.