CT image reconstruction method for small amount of projection data and differential equation solution method by neural network

通过使用微分方程的平方残差作为神经网络的目标函数，我们可以求解微分方程，因为网络本身在训练后成为解决方案。将该方法应用于Navier-Stokes方程、Poisson方程、Lorenz方程等，得到了满意的结果。我们发现，类似的方法适用于CT图像重建问题与少量的投影数据。将其应用于求解积分方程的CT图像重建问题，得到了满意的结果。通过推广目标函数的形式，这种方法可以应用于非常广泛的问题。用微分方程、积分方程和代数方程的残差平方乘以适当的惩罚系数来定义目标函数。由于多层神经网络具有良好的表达能力，因此可以比较容易地解决各种问题。其中一个重要的应用就是资料同化问题的求解，这在气象和海洋数值模拟领域中具有重要的意义。我们还进行了资料同化的模式数值试验，取得了满意的结果。该方法同样适用于各种科学和工程研究中出现的广义Abel反演，并与传统的标准方法进行了比较.作为CT图像重建的数学基础，我们研究了Radon逆变换。对于衰变Radon变换，我们导出了一个逆公式，并从数学上和数值上证明了这个逆公式。研究了无限区域上的Poisson方程和Helmholtz方程。我们引入了一个人工边界条件，并从数学和数值上研究了该条件的有效性。

By using squared residuals of a differential equation for the object function of a neural network we can solve the differential equation as the network itself becomes the solution after training. We investigated this method by applying it to Navier-Stokes equation, Poisson equation, Lorenz equation and so on, and obtained satisfactory results. We found that similar method is applied to CT image reconstruction problem with small amount of projection data. We applied it to model CT image reconstruction problem where an integral equation is solved and obtained satisfactory results. By generalizing the form of the object function this method can be applied to very wide range of problems. Defining the object function by the squared residuals of differential equations, integral equations, and algebraic equations multiplied by some appropriate penalty coefficients various kinds of problems can be solved owing to the excellent expressivity of the multi-layer neural network comparatively easily. One of the important application is the solution of data assimilation problem which is very important in the field of the meteorological and oceanological numerical simulations. We have also performed model numerical experiment of the data assimilation and obtained satisfactory results. This method is also applicable to the generalized Abel inversion which appears in various scientific and engineering researches.For comparing the new method with the conventional standard methods we studied these methods. As the mathematical basis of the CT image reconstruction we studied the inverse Radon transform. As for the decay Radon transform we derived an inverse formula, which we proved mathematically and numerically. We studied the Poisson equation and the Helmholtz equation in the infinite domain. We introduced an artificial boundary condition and studied the validity of the condition mathematically and numerically.