基于局部观测数据的一维热传导方程系数与初值同时反演问题的数值计算

利用局部观测数据研究了一维热传导方程系数与初值同时反演问题。首先利用函数代换及热传导方程解的解析延拓性，证明了基于局部观测数据下系数与初值同时反演问题的唯一性；然后基于深度神经网络，构造了同时反演问题的反演算法；最后通过构造具有解析解和不具有解析解的数值算例，验证了反演算法的有效性。

NUMERICAL SOLUTION OF SIMULTANEOUS IDENTIFICATION OF COEFFICIENT AND COEFFICIENT AND INITIAL VALUE FOR ONE DIMENSIONAL HEAT CONDUCTION EQUATION BASED ON LOCAL OBSERVATION DATA

The simultaneous inversion of coefficient and initial values for one-dimensional heat conduction equation is studied using local observation data. Firstly, the uniqueness of the simultaneous identification of coefficient and initial values is proved by using function substitution and analytic continuation technique of the solutions of parabolic equation based on the local observation data. Then, based on deep neural network, the inversion algorithm of simultaneous inversion problem is constructed. Finally, numerical examples with and without analytical solutions are constructed to verify the effectiveness of the inversion algorithm.