The ODYSEA (Ocean DYnamics and Surface Exchange with the Atmosphere) mission will provide simultaneous two-dimensional measurements of currents and winds for the first time. According to the ODYSEA radar concept, with a high incidence angle, current noise is primarily driven by backscattered power, which is triggered by wind speed. Therefore, random noise will affect the quality of observations. In low wind conditions, the absence of surface roughness increases the noise level considerably, to the point where the measurement becomes unusable, as the error can exceed 3 m/s at 5 km posting compared to mean current amplitudes of tens of cm/s. Winds higher than 7.5 m/s enable current measurements at 5 km posting with an RMS accuracy below 50 cm/s, but derivatives of currents will amplify noise, hampering the understanding of ocean dynamics and the interaction between the ocean and the atmosphere. In this context, this study shows the advantages and limitations of using noise-reduction algorithms. A convolutional neural network, a UNet inspired by the work of the SWOT (Surface Water and Ocean Topography) mission, is trained and tested on simulated radial velocities that are representative of the global ocean. The results are compared with those of classical smoothing: an Adaptive Gaussian Smoother whose filtering transfer function is optimized based on local wind speed (e.g., more smoothing in regions of low wind). The UNet outperforms the kernel smoother everywhere with our simulated dataset, especially in low wind conditions (SNR << 1) where the smoother essentially removes all velocities whereas the UNet mitigates random noise while preserving most of the signal of interest. Error is reduced by a factor of 30 and structures down to 30 km are reconstructed accurately. The UNet also enables the reconstruction of the main eddies and fronts in the relative vorticity field. It shows good robustness and stability in new scenarios.