Deep eutectic solvents (DESs) are gaining recognition as environmentally friendly solvent alternatives for diverse chemical processes. Yet, designing DESs tailored to specific applications is a resource-intensive task, which requires an accurate estimation of their physicochemical properties. Among them, viscosity is crucial, as it often dictates a DES's suitability as a solvent. In this study, an artificial neural network (ANN) is introduced to accurately describe the viscosity of DESs and their mixtures with cosolvents. The ANN utilizes molecular parameters derived from σ-profiles, computed using the conductor-like screening model for the real solvent segment activity coefficient (COSMO-SAC). The data set comprises 1891 experimental viscosity measurements for 48 DESs based on choline chloride, encompassing 279 different compositions, along with 1618 data points of DES mixtures with cosolvents as water, methanol, isopropanol, and dimethyl sulfoxide, covering a wide range of viscosity measurements from 0.3862 to 4722 mPa s. The optimal ANN structure for describing the logarithmic viscosity of DESs is configured as 9-19-16-1, achieving an overall average absolute relative deviation of 1.6031%. More importantly, the ANN shows a remarkable extrapolation capacity, as it is capable of predicting the viscosity of systems including solvents (ethanol) and hydrogen bond donors (2,3-butanediol) not considered in the training. The ANN model also demonstrates an extensive applicability domain, covering 94.17% of the entire database. These achievements represent a significant step forward in developing robust, open source, and highly accurate models for DESs using molecular descriptors.