Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization problem. The Lipchitz continuous of the gradient of cost function is proved based on the dual equation. In order to solve this optimization problem, this paper presents a modified conjugate gradient algorithm, and the global convergence of which is analyzed. The results of the simulation experiment show that the modified conjugate gradient algorithm obviously reduces the iterative number and running time. Due to the ill-posedness of the identification of boundary condition, this paper combines regularization method with the modified conjugate gradient algorithm. The simulation experiment illustrates that regularization method can eliminate the ill-posedness of this problem. Finally, the experimental data of a steel plant illustrate the validity of this paper’s method.

用非线性偏微分方程（PDE）描述了坯料的凝固传热过程。由于不良的生产环境，这种非线性PDE的边界条件很难确定。因此，考虑了二维非线性PDE边界条件的确定。本文将边界条件的识别转化为PDE优化问题。基于对偶方程证明了成本函数梯度的Lipchitz连续性。为了解决这一优化问题，本文提出了一种改进的共轭梯度算法，并对其全局收敛性进行了分析。仿真实验结果表明，改进的共轭梯度算法明显减少了迭代次数和运行时间。由于边界条件识别的不适性，本文将正则化方法与改进的共轭梯度算法相结合。仿真实验表明，正则化方法可以消除该问题的不适性。最后，某钢铁厂的实验数据证明了该方法的有效性。