The High-Altitude Water-Cherenkov (HAWC) experiment is a TeV $\gamma$-ray observatory located 4100 m above sea level on the Sierra Negra mountain in Puebla, Mexico. The detector consists of 300 water-filled tanks, each instrumented with 4 photomultiplier tubes that utilize the water-Cherenkov technique to detect atmospheric air showers produced by cosmic $\gamma$ rays. Construction of HAWC was completed in March of 2015. The experiment's wide instantaneous field of view (2 sr) and high duty cycle (> 95%) make it a powerful survey instrument sensitive to pulsars, supernova remnants, and other $\gamma$-ray sources. The mechanisms of particle acceleration at these sources can be studied by analyzing their high-energy spectra. To this end, we have developed an event-by-event energy-reconstruction algorithm using an artificial neural network to estimate energies of primary $\gamma$ rays at HAWC. We will present the details of this technique and its performance as well as the current progress toward using it to measure energy spectra of $\gamma$-ray sources.