Additive manufacturing (AM) enables the fabrication of objects using successive additions of mass and energy. In this paper, we explore the use of analytic solutions to model the thermal aspects of AM, in an effort to achieve high computational performance and enable “in the loop” use for feedback control of AM processes. It is shown that the utility of existing analytical solutions is limited due to their underlying assumption of a homogeneous semi-infinite domain. These solutions must, therefore, be enriched from their exact form in order to capture the relevant thermal physics associated with AM processes. Such enrichments include the handling of strong nonlinear variations in material properties, finite nonconvex solution domains, behavior of heat sources very near boundaries, and mass accretion coupled to the thermal problem. The enriched analytic solution method (EASM) is shown to produce results equivalent to those of numerical methods, which require six orders of magnitude greater computational effort. It is also shown that the EASM's computational performance is sufficient to enable AM process feedback control.