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

IF 1.9 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Jacques Francois Du Toit, R. Laubscher
{"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}
引用次数: 0

摘要

物理知情神经网络(PINN)是一类新的机器学习算法,能够在没有训练数据的情况下精确求解复杂偏微分方程(PDE)。通过引入一种新的流体模拟方法,PINN提供了解决以前难以解决的挑战的机会,例如不适定的PDE问题。PINN还可以以并行方式解决参数化问题,这导致了相关计算成本的有利缩放。PINN应用于解决流体动力学问题的全部潜力仍然未知,因为该方法仍处于早期开发阶段:许多问题仍有待解决,如训练动力学的数值刚度、湍流模拟方法的短缺以及围绕什么模型超参数表现最好的不确定性。在本文中,我们研究了PINN在层流和湍流状态下模拟主动脉跨瓣血流的准确性和效率,使用文献中的各种技术来提高PINN的模拟准确性。迄今为止,几乎没有发表过在没有训练数据的情况下使用PINN解决湍流的工作,因为事实证明,这种制度很困难。本文旨在通过提供一个此类应用的示例来解决文献中的这一空白。对模拟结果进行了讨论,并与有限体积法(FVM)的结果进行了比较。结果表明,PINN可以与层流的FVM解非常匹配,归一化最大速度和归一化最大压力误差分别低至5.74%和9.29%。湍流模拟是一个更大的挑战,归一化最大速度和归一化最大压力误差分别仅低至41.8%和113%。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Mathematical & Computational Applications
Mathematical & Computational Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-
自引率
10.50%
发文量
86
审稿时长
12 weeks
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信