The study of single-band quasar light curves poses two problems: inference of the power spectrum and interpolation of an irregularly sampled time series. A baseline approach to these tasks is to interpolate a time series with a Damped Random Walk (DRW) model, in which the spectrum is inferred using Maximum Likelihood Estimation (MLE). However, the DRW model does not describe the smoothness of the time series, whereas MLE faces many problems from the theory of optimization and computational math. In this work, we introduce a new stochastic model, that we call Convolved Damped Random Walk (CDRW). This model introduces a concept of smoothness to a Damped Random Walk, which enables it to fully describe quasar spectra. Moreover, we introduce a new method of inference of Gaussian process parameters, which we call Neural inference. This method uses the powers of the state-of-the-art neural networks to improve the conventional MLE inference technique. In our experiments, the Neural inference method results in significant improvement over the baseline MLE (RMSE: 0.318 → 0.205, 0.464 → 0.444). Moreover, the combination of both the CDRW model and the Neural inference significantly outperforms the baseline DRW and MLE in the interpolation of a typical quasar light curve (χ2: 0.333 → 0.998, 2.695 → 0.981).