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High-energy radiation from the Sun governs the behavior of Earth's upper atmosphere and such radiation from any planet-hosting star can drive the long-term evolution of a planetary atmosphere. However, much of this radiation is unobservable because of absorption by Earth's atmosphere and the interstellar medium. This motivates the identification of a proxy that can be readily observed from the ground. Here, we evaluate absorption in the near-infrared 1083 nm triplet line of neutral orthohelium as a proxy for extreme ultraviolet (EUV) emission in the 30.4 nm line of He II and 17.1 nm line of Fe IX from the Sun. We apply deep learning to model the non-linear relationships, training and validating the model on historical, contemporaneous images of the solar disk acquired in the triplet He I line by the ground-based SOLIS observatory and in the EUV by the NASA Solar Dynamics Observatory. The model is a fully-convolutional neural network (FCN) that incorporates spatial information and accounts for the projection of the spherical Sun to 2-d images. Using normalized target values, results indicate a median pixel-wise relative error of 20% and a mean disk-integrated flux error of 7% on a held-out test set. Qualitatively, the model learns the complex spatial correlations between He I absorption and EUV emission has a predictive ability superior to that of a pixel-by-pixel model; it can also distinguish active regions from high-absorption filaments that do not result in EUV emission., Comment: Accepted to ApJ Letters