Spatial heterodyne spectroscopy is a new type of spatially modulated Fourier interference spectroscopy technology that can achieve ultra-high spectral resolution within a determined spectral range. It has unique advantages in atmospheric remote sensing, astronomical observation, and ore detection. Due to the complex detection environment and interference from electronic components themselves, spatial heterodyne spectrometers may also be affected by noise during the detection process. With the continuous improvement of signal detection accuracy, there is an urgent need for new methods to reduce the impact of noise on the spectral information contained in spatial heterodyne interferograms. In recent years, deep neural networks have experienced rapid development and achieved good results in regression prediction of micro nano devices and image denoising. Therefore, we prepared a training set using the principle of spatial heterodyne spectroscopy to train the constructed deep neural network, and then used the trained network model to directly output the denoised spatial heterodyne spectra from the spatial heterodyne interferogram containing Gaussian noise. The results show that deep neural networks have excellent ability to extract denoised spatial heterodyne spectra from noise spatial heterodyne interferograms. This study provides a new and effective solution for denoising spatial heterodyne interferometric spectral information.