Dimensionality reduction and physics-based manifold learning for parametric models in biomechanics and tissue engineering

A. Muixí, A. Garcia-Gonzalez, S. Zlotnik, P. Díez
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Abstract

This work aims at describing dimensionality reduction methods, particularizing in Principal Component Analysis (PCA), the nonlinear version kernel Principal Component Analysis (kPCA) [1], and their potential application to data-assisted Credible models in biomechanics and tissue engineering. These methodologies are intended to discover the low dimensional manifold where an input physical data set lives. Reducing the dimensionality of a complex physical system is a potential tool towards real time Credible and accurate parametric models and patient-specific simulations. In this direction, the Proper Orthogonal Decomposition (POD) combines PCA with a reduced basis approach to reduce the number of degrees of freedom in parametric boundary value problems. Additionally, for systems whose solutions belong to nonlinear manifolds, kernel Proper Orthogonal Decomposition (kPOD) uses kPCA reduction to find a solution of the problem. The main features of kPOD are the use of local approximations, the possibility of enriching the reduced space with quadratic elements, the use of ad-hoc kernels that include previous knowledge of the input data, and the idea of using an iterative algorithm that explores the Voronoi diagram of the snapshots in the reduced space [2]. Besides, dimensionality reduction in combination with surrogate modelling aims at finding initial (and accurate) approximations of parametric systems without physics involved. All presented methodologies are shown to be strong tools in several fields. To show the potential of those techniques, here we present several examples of application in the biomechanical field, such as advection diffusion in scaffolds for tissue engineering, and vascular biomechanics
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生物力学和组织工程中参数化模型的降维和基于物理的流形学习
本工作旨在描述降维方法,特别是主成分分析(PCA)、非线性核主成分分析(kPCA)[1],以及它们在生物力学和组织工程中数据辅助可信模型中的潜在应用。这些方法旨在发现输入物理数据集所在的低维流形。降低复杂物理系统的维数是实现实时、可靠和准确的参数模型和特定患者模拟的潜在工具。在这个方向上,适当正交分解(POD)将主成分分析与降基方法相结合,减少了参数边值问题的自由度。此外,对于解属于非线性流形的系统,核固有正交分解(kPOD)使用kPCA约简来寻找问题的解。kPOD的主要特征是使用局部近似,用二次元丰富约简空间的可能性,使用包含输入数据先前知识的特别核,以及使用迭代算法探索约简空间中快照的Voronoi图的想法[2]。此外,降维与代理建模相结合的目的是在不涉及物理的情况下找到参数系统的初始(和准确)近似。所有提出的方法都被证明是几个领域的强大工具。为了展示这些技术的潜力,我们在这里给出几个应用在生物力学领域的例子,如组织工程支架的平流扩散和血管生物力学
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