A Genetic Programming Approach To Partial Differential Equation Inverse Problems

It is well known that many problems founded in physics, mechanics and engineer and so on can be described with partial differential equation. With the development of society, the actual demand of each field come from producing, living (such as, seeking the change law of materials lying in the place that can't be reached; designing the products according to the specific function; making the procedure according to a certain purpose; hoping to get a certain new material in industrial production etc.) has promoted rapid development against the inverse problems of partial differential equation.In recent years, different disciplines domain has developed some methods to solve the inverse problems, but these methods often all have certain limitation. Some require there are special forms in the equation; some are the restriction on geometry is strong. PST and perturbation method are more effective number value methods, they belong to linear or simulation linear inversion method, but this kind of method depends on the choosing of initial model strongly.Evolutionary computation (EC) is an intelligence algorithm that learns from the evolutionary process in the nature. EC employs a coding technology and some genetic operations. Under the pressure of selection, which means "fits survive", the algorithm can produce an optimal solution. Because EC is simple and it seldom needs any additional information about the problem, EC becomes a general solver of challenge problems.This paper is to propose essential theory based on partial differential equations inverse problem. Due to its intrinsic ill-posed nature, we must use stabilizing measures to deal with regularization. Moreover, mathematical methods have much chance to get stuck at local minima, unless a good starting point is available. Clearly classical methods operate locally and are not intrinsically parallel. So we bring EAs into partial differential equations inverse problem due to its character. In this paper we solve the continuous and disconnected elliptic partial differential equation parameter identificationproblems separately using GP based on the parameter estimation and PTGP. The Numerical results show that these two kinds of algorithms are feasible andeffective.