José-Luis Llaguno-Roque, R. Barrientos-Martínez, H. Acosta-Mesa, T. Romo-González, E. Mezura-Montes
Breast cancer has become a global health problem, ranking first in incidences and fifth in mortality in women around the world. In Mexico, the first cause of death in women is breast cancer. This work uses deep learning techniques to discriminate between healthy and breast cancer patients, based on the banding patterns obtained from the Western Blot strip images of the autoantibody response to antigens of the T47D tumor line. The reaction of antibodies to tumor antigens occurs early in the process of tumorigenesis, years before clinical symptoms. One of the main challenges in deep learning is the design of the architecture of the convolutional neural network. Neuroevolution has been used to support this and has produced highly competitive results. It is proposed that neuroevolve convolutional neural networks (CNN) find an optimal architecture to achieve competitive ranking, taking Western Blot images as input. The CNN obtained reached 90.67% accuracy, 90.71% recall, 95.34% specificity, and 90.69% precision in classifying three different classes (healthy, benign breast pathology, and breast cancer).
{"title":"Neuroevolution of Convolutional Neural Networks for Breast Cancer Diagnosis Using Western Blot Strips","authors":"José-Luis Llaguno-Roque, R. Barrientos-Martínez, H. Acosta-Mesa, T. Romo-González, E. Mezura-Montes","doi":"10.3390/mca28030072","DOIUrl":"https://doi.org/10.3390/mca28030072","url":null,"abstract":"Breast cancer has become a global health problem, ranking first in incidences and fifth in mortality in women around the world. In Mexico, the first cause of death in women is breast cancer. This work uses deep learning techniques to discriminate between healthy and breast cancer patients, based on the banding patterns obtained from the Western Blot strip images of the autoantibody response to antigens of the T47D tumor line. The reaction of antibodies to tumor antigens occurs early in the process of tumorigenesis, years before clinical symptoms. One of the main challenges in deep learning is the design of the architecture of the convolutional neural network. Neuroevolution has been used to support this and has produced highly competitive results. It is proposed that neuroevolve convolutional neural networks (CNN) find an optimal architecture to achieve competitive ranking, taking Western Blot images as input. The CNN obtained reached 90.67% accuracy, 90.71% recall, 95.34% specificity, and 90.69% precision in classifying three different classes (healthy, benign breast pathology, and breast cancer).","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41960318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated with Sierpiński lattices and infinite Sierpiński gaskets and other post-critically finite self-similar sets.
{"title":"Sierpiński Fractals and the Dimension of Their Laplacian Spectrum","authors":"M. Pollicott, J. Slipantschuk","doi":"10.3390/mca28030070","DOIUrl":"https://doi.org/10.3390/mca28030070","url":null,"abstract":"We establish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated with Sierpiński lattices and infinite Sierpiński gaskets and other post-critically finite self-similar sets.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44102642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Robust and computationally efficient numeric algorithms are required to simulate the sintering process of complex ceramic components by means of the finite element method. This work focuses on a thermodynamically consistent sintering model capturing the effects of both, viscosity and elasticity, within the standard dissipative framework. In particular, the temporal integration of the model by means of several implicit first and second order accurate one step time integration methods is discussed. It is shown in terms of numerical experiments on the material point level that the first order schemes exhibit poor performance when compared to second order schemes. Further numerical experiments indicate that the results translate directly to finite element simulations.
{"title":"Numerical Aspects of a Continuum Sintering Model Formulated in the Standard Dissipative Framework","authors":"S. Stark","doi":"10.3390/mca28030069","DOIUrl":"https://doi.org/10.3390/mca28030069","url":null,"abstract":"Robust and computationally efficient numeric algorithms are required to simulate the sintering process of complex ceramic components by means of the finite element method. This work focuses on a thermodynamically consistent sintering model capturing the effects of both, viscosity and elasticity, within the standard dissipative framework. In particular, the temporal integration of the model by means of several implicit first and second order accurate one step time integration methods is discussed. It is shown in terms of numerical experiments on the material point level that the first order schemes exhibit poor performance when compared to second order schemes. Further numerical experiments indicate that the results translate directly to finite element simulations.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46218464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduced a comparison method for three explicitly defined intermediate encoding methods in generative design for two-dimensional soft robotic units. This study evaluates a conventional genetic algorithm with full access to removing elements from the design domain using an implicit random encoding layer, a Lindenmayer system encoding mimicking biological growth patterns and a compositional pattern producing network encoding for 2D pattern generation. The objective of the optimisation problem is to match the deformation of a single actuator unit with a desired target shape, specifically uni-axial elongation, under internal pressure. The study results suggest that the Lindenmayer system encoding generates candidate units with fewer function evaluations than the traditional implicitly encoded genetic algorithm. However, the distribution of constraint and internal energy is similar to that of the random encoding, and the Lindenmayer system encoding produces a less diverse population of candidate units. In contrast, despite requiring more function evaluations than the Lindenmayer System encoding, the Compositional Pattern Producing Network encoding produces a similar diversity of candidate units. Overall, the Compositional Pattern Producing Network encoding results in a proportionally higher number of high-performing units than the random or Lindenmayer system encoding, making it a viable alternative to a conventional monolithic approach. The results suggest that the compositional pattern producing network encoding may be a promising approach for designing soft robotic actuators with desirable performance characteristics.
{"title":"Intermediate Encoding Layers for the Generative Design of 2D Soft Robot Actuators: A Comparison of CPPN’s, L-Systems and Random Generation","authors":"M. Venter, Naudé Thomas Conradie","doi":"10.3390/mca28030068","DOIUrl":"https://doi.org/10.3390/mca28030068","url":null,"abstract":"This paper introduced a comparison method for three explicitly defined intermediate encoding methods in generative design for two-dimensional soft robotic units. This study evaluates a conventional genetic algorithm with full access to removing elements from the design domain using an implicit random encoding layer, a Lindenmayer system encoding mimicking biological growth patterns and a compositional pattern producing network encoding for 2D pattern generation. The objective of the optimisation problem is to match the deformation of a single actuator unit with a desired target shape, specifically uni-axial elongation, under internal pressure. The study results suggest that the Lindenmayer system encoding generates candidate units with fewer function evaluations than the traditional implicitly encoded genetic algorithm. However, the distribution of constraint and internal energy is similar to that of the random encoding, and the Lindenmayer system encoding produces a less diverse population of candidate units. In contrast, despite requiring more function evaluations than the Lindenmayer System encoding, the Compositional Pattern Producing Network encoding produces a similar diversity of candidate units. Overall, the Compositional Pattern Producing Network encoding results in a proportionally higher number of high-performing units than the random or Lindenmayer system encoding, making it a viable alternative to a conventional monolithic approach. The results suggest that the compositional pattern producing network encoding may be a promising approach for designing soft robotic actuators with desirable performance characteristics.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48058150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Luis Víctor Maidana Benítez, Melisa María Rosa Villamayor Paredes, José Colbes, César F. Bogado-Martínez, B. Barán, D. Pinto-Roa
This paper addresses serialized approaches of the routing, modulation level, and spectrum assignment (RMLSA) problem in elastic optical networks, using multiple sequential sub-sets of requests, under Integer Linear Programming (ILP). The literature has reported two-stage serial optimization methods referred to as RML+SA, which retain computational efficiency when the problem grows, compared to the classical one-stage RMLSA optimization approach. However, there still remain numerous issues in terms of the spectrum used that can be improved when compared to the RMLSA solution. Consequently, this paper proposes RML+SA solutions considering multiple sequential sub-sets of requests, split traffic flow, as well as path-oriented and link-oriented routing models. Simulation results on different test scenarios determine that: (a) the multiple sequential sub-sets of request-based models improve computation time without worsening the spectrum usage when compared to just one set of requests optimization approaches, (b) divisible traffic flow approaches show promise in cases where the number of request sub-sets is low compared to the non-divisible counterpart, and (c) path-oriented routing succeeds in improving the used spectrum by increasing the number of candidate routes compared to link-oriented routing.
{"title":"Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple Sub-Sets of Requests Based on Integer Linear Programming","authors":"Luis Víctor Maidana Benítez, Melisa María Rosa Villamayor Paredes, José Colbes, César F. Bogado-Martínez, B. Barán, D. Pinto-Roa","doi":"10.3390/mca28030067","DOIUrl":"https://doi.org/10.3390/mca28030067","url":null,"abstract":"This paper addresses serialized approaches of the routing, modulation level, and spectrum assignment (RMLSA) problem in elastic optical networks, using multiple sequential sub-sets of requests, under Integer Linear Programming (ILP). The literature has reported two-stage serial optimization methods referred to as RML+SA, which retain computational efficiency when the problem grows, compared to the classical one-stage RMLSA optimization approach. However, there still remain numerous issues in terms of the spectrum used that can be improved when compared to the RMLSA solution. Consequently, this paper proposes RML+SA solutions considering multiple sequential sub-sets of requests, split traffic flow, as well as path-oriented and link-oriented routing models. Simulation results on different test scenarios determine that: (a) the multiple sequential sub-sets of request-based models improve computation time without worsening the spectrum usage when compared to just one set of requests optimization approaches, (b) divisible traffic flow approaches show promise in cases where the number of request sub-sets is low compared to the non-divisible counterpart, and (c) path-oriented routing succeeds in improving the used spectrum by increasing the number of candidate routes compared to link-oriented routing.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43420132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Images or paintings with homogeneous colors may appear dull to the naked eye; however, there may be numerous details in the image that are expressed through subtle changes in color. This manuscript introduces a novel approach that can uncover these concealed details via a transformation that increases the distance between adjacent pixels, ultimately leading to a newly modified version of the input image. We chose the artworks of Mark Rothko—famous for their simplicity and limited color palette—as a case study. Our approach offers a different perspective, leading to the discovery of either accidental or deliberate clusters of colors. Our method is based on the quaternion ring, wherein a suitable multiplication can be used to boost the color difference between neighboring pixels, thereby unveiling new details in the image. The quality of the transformation between the original image and the resultant versions can be measured by the ratio between the number of connected components in the original image (m) and the number of connected components in the output versions (n), which usually satisfies nm≫1. Although this procedure has been employed as a case study for artworks, it can be applied to any type of image with a similar simplicity and limited color palette.
{"title":"Applying Quaternions to Recognize Hidden Details in Images: Rothko as a Case Study","authors":"Adam Aharony, Ron Hindi, Maor Valdman, Shai Gul","doi":"10.3390/mca28030066","DOIUrl":"https://doi.org/10.3390/mca28030066","url":null,"abstract":"Images or paintings with homogeneous colors may appear dull to the naked eye; however, there may be numerous details in the image that are expressed through subtle changes in color. This manuscript introduces a novel approach that can uncover these concealed details via a transformation that increases the distance between adjacent pixels, ultimately leading to a newly modified version of the input image. We chose the artworks of Mark Rothko—famous for their simplicity and limited color palette—as a case study. Our approach offers a different perspective, leading to the discovery of either accidental or deliberate clusters of colors. Our method is based on the quaternion ring, wherein a suitable multiplication can be used to boost the color difference between neighboring pixels, thereby unveiling new details in the image. The quality of the transformation between the original image and the resultant versions can be measured by the ratio between the number of connected components in the original image (m) and the number of connected components in the output versions (n), which usually satisfies nm≫1. Although this procedure has been employed as a case study for artworks, it can be applied to any type of image with a similar simplicity and limited color palette.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47507597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a brief survey of selected results obtained using the “transversality method” developed for studying parametrized families of fractal sets and measures. We mostly focus on the early development of the theory, restricting ourselves to self-similar and self-conformal iterated function systems.
{"title":"Notes on the Transversality Method for Iterated Function Systems—A Survey","authors":"B. Solomyak","doi":"10.3390/mca28030065","DOIUrl":"https://doi.org/10.3390/mca28030065","url":null,"abstract":"This is a brief survey of selected results obtained using the “transversality method” developed for studying parametrized families of fractal sets and measures. We mostly focus on the early development of the theory, restricting ourselves to self-similar and self-conformal iterated function systems.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44529841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Khanyisani Makhanya, S. Connell, M. Bhamjee, N. Martinson
Pulmonary diseases are a leading cause of illness and disability globally. While having access to hospitals or specialist clinics for investigations is currently the usual way to characterize the patient’s condition, access to medical services is restricted in less resourced settings. We posit that pulmonary disease may impact on vocalization which could aid in characterizing a pulmonary condition. We therefore propose a new method to diagnose pulmonary disease analyzing the vocal and cough changes of a patient. Computational fluid dynamics holds immense potential for assessing the flow-induced acoustics in the lungs. The aim of this study is to investigate the potential of flow-induced vocal-, cough-, and lung-generated acoustics to diagnose lung conditions using computational fluid dynamics methods. In this study, pneumonia is the model disease which is studied. The hypothesis is that using a computational fluid dynamics model for assessing the flow-induced acoustics will accurately represent the flow-induced acoustics for healthy and infected lungs and that possible modeled difference in fluid and acoustic behavior between these pathologies will be tested and described. Computational fluid dynamics and a lung geometry will be used to simulate the flow distribution and obtain the acoustics for the different scenarios. The results suggest that it is possible to determine the difference in vocalization between healthy lungs and those with pneumonia, using computational fluid dynamics, as the flow patterns and acoustics differ. Our results suggest there is potential for computational fluid dynamics to enhance understanding of flow-induced acoustics that could be characteristic of different lung pathologies. Such simulations could be repeated using machine learning with the final objective to use telemedicine to triage or diagnose patients with respiratory illness remotely.
{"title":"The Use of Computational Fluid Dynamics for Assessing Flow-Induced Acoustics to Diagnose Lung Conditions","authors":"Khanyisani Makhanya, S. Connell, M. Bhamjee, N. Martinson","doi":"10.3390/mca28030064","DOIUrl":"https://doi.org/10.3390/mca28030064","url":null,"abstract":"Pulmonary diseases are a leading cause of illness and disability globally. While having access to hospitals or specialist clinics for investigations is currently the usual way to characterize the patient’s condition, access to medical services is restricted in less resourced settings. We posit that pulmonary disease may impact on vocalization which could aid in characterizing a pulmonary condition. We therefore propose a new method to diagnose pulmonary disease analyzing the vocal and cough changes of a patient. Computational fluid dynamics holds immense potential for assessing the flow-induced acoustics in the lungs. The aim of this study is to investigate the potential of flow-induced vocal-, cough-, and lung-generated acoustics to diagnose lung conditions using computational fluid dynamics methods. In this study, pneumonia is the model disease which is studied. The hypothesis is that using a computational fluid dynamics model for assessing the flow-induced acoustics will accurately represent the flow-induced acoustics for healthy and infected lungs and that possible modeled difference in fluid and acoustic behavior between these pathologies will be tested and described. Computational fluid dynamics and a lung geometry will be used to simulate the flow distribution and obtain the acoustics for the different scenarios. The results suggest that it is possible to determine the difference in vocalization between healthy lungs and those with pneumonia, using computational fluid dynamics, as the flow patterns and acoustics differ. Our results suggest there is potential for computational fluid dynamics to enhance understanding of flow-induced acoustics that could be characteristic of different lung pathologies. Such simulations could be repeated using machine learning with the final objective to use telemedicine to triage or diagnose patients with respiratory illness remotely.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43078812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The distribution of the sum of negative binomial random variables has a special role in insurance mathematics, actuarial sciences, and ecology. Two methods to estimate this distribution have been published: a finite-sum exact expression and a series expression by convolution. We compare both methods, as well as a new normalized saddlepoint approximation, and normal and single distribution negative binomial approximations. We show that the exact series expression used lots of memory when the number of random variables was high (>7). The normalized saddlepoint approximation gives an output with a high relative error (around 3–5%), which can be a problem in some situations. The convolution method is a good compromise for applied practitioners, considering the amount of memory used, the computing time, and the precision of the estimates. However, a simplistic implementation of the algorithm could produce incorrect results due to the non-monotony of the convergence rate. The tolerance limit must be chosen depending on the expected magnitude order of the estimate, for which we used the answer generated by the saddlepoint approximation. Finally, the normal and negative binomial approximations should not be used, as they produced outputs with a very low accuracy.
{"title":"Computation of the Distribution of the Sum of Independent Negative Binomial Random Variables","authors":"M. Girondot, J. Barry","doi":"10.3390/mca28030063","DOIUrl":"https://doi.org/10.3390/mca28030063","url":null,"abstract":"The distribution of the sum of negative binomial random variables has a special role in insurance mathematics, actuarial sciences, and ecology. Two methods to estimate this distribution have been published: a finite-sum exact expression and a series expression by convolution. We compare both methods, as well as a new normalized saddlepoint approximation, and normal and single distribution negative binomial approximations. We show that the exact series expression used lots of memory when the number of random variables was high (>7). The normalized saddlepoint approximation gives an output with a high relative error (around 3–5%), which can be a problem in some situations. The convolution method is a good compromise for applied practitioners, considering the amount of memory used, the computing time, and the precision of the estimates. However, a simplistic implementation of the algorithm could produce incorrect results due to the non-monotony of the convergence rate. The tolerance limit must be chosen depending on the expected magnitude order of the estimate, for which we used the answer generated by the saddlepoint approximation. Finally, the normal and negative binomial approximations should not be used, as they produced outputs with a very low accuracy.","PeriodicalId":53224,"journal":{"name":"Mathematical & Computational Applications","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45275097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"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":"https://doi.org/10.3390/mca28020062","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":1.9,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48080534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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