Task-based measures of image quality (IQ) quantify the ability of an observer to perform a specific task. Such measures are commonly employed for assessing and optimizing medical imaging systems. In binary signal detection tasks, the Bayesian ideal observer (IO) sets an upper performance limit. However, the IO test statistic is generally intractable to compute when the log-likelihood ratio depends non-linearly on the measurement data. In such cases, the Hotelling observer (HO), which is the optimal linear observer, can be employed. However, traditional implementations of the HO require estimation and inversion of covariance matrices; for large images this can be computationally burdensome or even intractable. In this work, we describe a novel supervised learning- based method that employs artificial neural networks (ANNs) for estimating the HO test statistic and does not require estimation or inversion of covariance matrices. A signal-known-exactly and background-known-exactly (SKE/BKE) signal detection task is considered. The receiver operating characteristic (ROC) curve and Hotelling template corresponding to the proposed method are compared to the corresponding analytical solutions.