Deep learning algorithms have been successfully applied to numerically solve linear Kolmogorov partial differential equations (PDEs). A recent research shows that the empirical risk minimization~(ERM) over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of linear Kolmogorov PDEs with bounded initial functions. However, the initial functions may be unbounded in many applications such as the Black Scholes PDEs in pricing call options. In this paper, we extend this result to the cases involving unbounded initial functions. We prove that for $d$-dimensional linear Kolmogorov PDEs with unbounded initial functions, under suitable assumptions, the number of training data and the size of the artificial neural network required to achieve an accuracy $\varepsilon$ for the ERM grow polynomially in both $d$ and $\varepsilon^{-1}$. Moreover, we verify that the required assumptions hold for Black-Scholes PDEs and heat equations which are two important cases of linear Kolmogorov PDEs.

翻译：深度学习算法已被成功应用于线性Kolmogorov偏微分方程(PDEs)的数值求解。近期研究表明，对于具有有界初始函数的线性Kolmogorov PDEs，深度人工神经网络上的经验风险最小化(ERM)在数值逼近中克服了维数灾难。然而，在许多应用中（如期权定价中的Black-Scholes PDEs），初始函数可能无界。本文将此结果推广至涉及无界初始函数的情形。我们证明：对于具有无界初始函数的$d$维线性Kolmogorov PDEs，在适当假设下，为实现ERM精度$\varepsilon$所需的训练数据量与人工神经网络规模均关于$d$和$\varepsilon^{-1}$多项式增长。此外，我们验证了上述假设在Black-Scholes PDEs和热方程（线性Kolmogorov PDEs的两个重要特例）中成立。