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A deep understanding of the neural architecture of mental function should enable the accurate prediction of a specific pattern of brain activity for any psychological task, based only on the cognitive functions known to be engaged by that task. Encoding models (EMs), which predict neural responses from known features (e.g., stimulus properties), have succeeded in circumscribed domains (e.g., visual neuroscience), but implementing domain-general EMs that predict brain-wide activity for arbitrary tasks has been limited mainly by availability of datasets that 1) sufficiently span a large space of psychological functions, and 2) are sufficiently annotated with such functions to allow robust EM specification. We examine the use of EMs based on a formal specification of psychological function, to predict cortical activation patterns across a broad range of tasks. We utilized the Multi-Domain Task Battery, a dataset in which 24 subjects completed 32 ten-minute fMRI scans, switching tasks every 35 s and engaging in 44 total conditions of diverse psychological manipulations. Conditions were annotated by a group of experts using the Cognitive Atlas ontology to identify putatively engaged functions, and region-wise cognitive EMs (CEMs) were fit, for individual subjects, on neocortical responses. We found that CEMs predicted cortical activation maps of held-out tasks with high accuracy, outperforming a permutation-based null model while approaching the noise ceiling of the data, without being driven solely by either cognitive or perceptual-motor features. Hierarchical clustering on the similarity structure of CEM generalization errors revealed relationships amongst psychological functions. Spatial distributions of feature importances systematically overlapped with large-scale resting-state functional networks (RSNs), supporting the hypothesis of functional specialization within RSNs while grounding their function in an interpretable data-driven manner. Our implementation and validation of CEMs provides a proof of principle for the utility of formal ontologies in cognitive neuroscience and motivates the use of CEMs in the further testing of cognitive theories.