求解反问题的一个新方法:模型求解与范例学习结合

发布时间：2018-03-09 01:22

本文选题：反问题　切入点：范例学习　出处：《中国科学:数学》2017年10期 　论文类型：期刊论文

【摘要】：反问题存在于科学、工程与技术等广泛领域,是数学问题的基本形式之一.传统的反问题求解基于求解与问题相关的数学模型(如一个偏微分方程组和变分问题等),无论是建模还是模型求解都回避不了大量近似,从而很难获得满意的准确解,被公认是困难问题.然而,我们注意到,对很多反问题(如图像处理)而言,除模型之外,还往往存在大量的范例.本文提出将模型求解与范例学习相结合的反问题求解新思路.以压缩感知核磁共振成像(CS-MRI)为例,说明如何将压缩感知模型的模型求解与基于范例学习的深度学习相融合,以形成求解CS-MRI问题的一个全新方法.应用表明,基于模型与基于数据相结合的这一新方法不仅可行、高效,而且具有很强的可推广性.
[Abstract]:Inverse problems exist in a wide range of fields, such as science, engineering and technology, Traditional inverse problem solving is based on solving mathematical models related to the problem (such as a partial differential equation system and variational problems etc.) both modeling and model solving can not avoid a large number of approximations. It is difficult to get a satisfactory and accurate solution, which is recognized as a difficult problem. However, we note that for many inverse problems (such as image processing), except for the model, In this paper, we propose a new idea of inverse problem solving which combines model solving with case learning. Take compressed perceptual magnetic resonance imaging (CS-MRI) as an example, This paper explains how to combine the model solving of compressed perceptual model with the deep learning based on case learning to form a new method for solving CS-MRI problem. The application shows that the new method based on model and data is not only feasible. High efficiency and strong extensibility.
【作者单位】：西安交通大学数学与统计学院;
【基金】：国家自然科学基金(批准号:91330204)资助项目
【分类号】：O241.6

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