  We show that Weyl superconductivity can be induced by finite supercurrent in
noncentrosymmetric spin-orbit-coupled superconductors with line nodes. We
introduce a three-dimensional tight-binding model of a tetragonal
superconductor in a $D+p$-wave pairing state with a finite center-of-mass
momentum, and elucidate that a line-nodal to point-nodal spectral transition
occurs by applying an infinitesimal supercurrent. We also clarify that the
higher-order effect in spin-orbit coupling is particularly important for this
phenomenon. The point nodes are protected by topologically nontrivial Weyl
charges, and therefore gapless arc states appear on the surface of the
superconductor. Furthermore, both the positions and the Weyl charges of the
point nodes depend on the direction of the current. In addition, a quantized
Berry phase defined on high-symmetry planes characterizes the Weyl nodes when
the in-plane supercurrent is considered. Our proposition paves a new way for
controlling the superconducting gap structures by using an external field.
