  A model-independent test of the cosmic curvature parameter $\Omega_k$ is very
important in cosmology. In order to estimate cosmic curvature from cosmological
probes like standard candles, one has to be able to measure the luminosity
distance $D_L(z)$, it's derivative with respect to redshift $D'_L(z)$ and
independently know the expansion rate $H(z)$ at the same redshift. In this
paper, we study how such an idea could be implemented with the future
generation of space-based DECi-hertz Interferometer Gravitational-wave
Observatory (DECIGO), in combination with cosmic chronometers providing
cosmology-independent $H(z)$ data. Our results show that for the Hubble diagram
of simulated DECIGO data acting as a new type of standard siren, it would be
able to constrain cosmic curvature with the precision of $\Delta \Omega_k=
0.09$ with the currently available sample of 31 measurements of Hubble
parameters. In the framework of the third generation ground-based gravitational
wave detectors, the spatial curvature is constrained to be $\Delta\Omega_k=
0.13$ for Einstein Telescope (ET). More interestingly, compared to other
approaches aiming for model-independent estimations of spatial curvature, our
analysis also achieves the reconstruction of the evolution of $\Omega_k(z)$, in
the framework of a model-independent method of Gaussian processes (GP) without
assuming a specific form. Therefore, one can expect that the newly emerged
gravitational wave astronomy can become useful in local measurements of cosmic
curvature using distant sources.
