Empirical Risk Minimization over Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Linear Kolmogorov Partial Differential Equations with Unbounded Initial Functions

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.