  Deep learning algorithms have been successfully applied to numerically solve
linear Kolmogorov partial differential equations~(PDEs). A recent research
shows that if the initial functions are bounded, the empirical risk
minimization (ERM) over clipped ReLU networks generalizes well for solving the
linear Kolmogorov PDE. In this paper, we propose to use a truncation technique
to extend the generalization results for polynomially growing initial
functions. Specifically, we prove that under an assumption, the sample size
required to achieve an generalization error within $\varepsilon$ with a
confidence level $\varrho$ grows polynomially in the size of the clipped neural
networks and $(\varepsilon^{-1},\varrho^{-1})$, which means that the curse of
dimensionality is broken. Moreover, we verify that the required assumptions
hold for Black-Scholes PDEs and heat equations which are two important cases of
linear Kolmogorov PDEs. For the approximation error, under certain assumptions,
we establish approximation results for clipped ReLU neural networks when
approximating the solution of Kolmogorov PDEs. Consequently, we establish that
the ERM over artificial neural networks indeed overcomes the curse of
dimensionality for a larger class of linear Kolmogorov PDEs.
