The Zwicky Transient Facility (ZTF), a state-of-the-art optical robotic sky survey, registers on the order of a million transient events - such as supernova explosions, changes in brightness of variable sources, or moving object detections - every clear night, generating real-time alerts. We present Alert-Classifying Artificial Intelligence (ACAI), an open-source deep-learning framework for the phenomenological classification of the ZTF alerts. ACAI uses a set of five binary classifiers to characterize objects, which in combination with the auxiliary/contextual event information available from alert brokers, provides a powerful tool for alert stream filtering tailored to different science cases, including early identification of supernova-like and anomalous transient events. We report on the performance of ACAI during the first months of deployment in a production setting.