  We present the Lyman-$\alpha$ Continuum Analysis Network (LyCAN), a
Convolutional Neural Network that predicts the unabsorbed quasar continuum
within the rest-frame wavelength range of $1040-1600$ Angstroms based on the
red side of the Lyman-$\alpha$ emission line ($1216-1600$ Angstroms). We
developed synthetic spectra based on a Gaussian Mixture Model representation of
Nonnegative Matrix Factorization (NMF) coefficients. These coefficients were
derived from high-resolution, low-redshift ($z<0.2$) Hubble Space
Telescope/Cosmic Origins Spectrograph quasar spectra. We supplemented this
COS-based synthetic sample with an equal number of DESI Year 5 mock spectra.
LyCAN performs extremely well on testing sets, achieving a median error in the
forest region of 1.5% on the DESI mock sample, 2.0% on the COS-based synthetic
sample, and 4.1% on the original COS spectra. LyCAN outperforms Principal
Component Analysis (PCA)- and NMF-based prediction methods using the same
training set by a factor of two or more. We predict the intrinsic continua of
83,635 DESI Year 1 spectra in the redshift range of $2.1 \leq z \leq 4.2$ and
perform an absolute measurement of the evolution of the effective optical
depth. This is the largest sample employed to measure the optical depth
evolution to date. We fit a power-law of the form $\tau(z) = \tau_0
(1+z)^\gamma$ to our measurements and find $\tau_0 = (2.46 \pm
0.14)\times10^{-3}$ and $\gamma = 3.62 \pm 0.04$. Our results show particular
agreement with high-resolution, ground-based observations around $z = 2$,
indicating that LyCAN is able to predict the quasar continuum in the forest
region with only spectral information outside the forest.
