  In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For a weighted oriented graph $D$, we give a lower bound for $\reg(I(D)^{k})$, if $V^+$ are sinks. If $D$ has an induced directed path $(x_i,x_j),(x_j,x_r) \in E(D)$ of length $2$ with $w(x_j)\geq 2$, then we show that $\reg(I(D)^{k})\leq \reg(I(D)^{k})$ for all $k\geq 2$. In particular, if $D$ is bipartite, then the above inequality holds for all $k\geq 2$. For any weighted oriented graph $D$, if $V^+$ are sink vertices, then we show that $\reg(I(D)^{k}) \leq \reg(I(D)^{k})$ with $k=2,3$. We further study when these regularities are equal. As a consequence, we give sharp upper bounds for regularity of symbolic powers of certain class of weighted oriented graphs. 
Regularity comparison of symbolic and ordinary powers of weighted   oriented graphs