抄録
Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3500000, indirect: -)
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.