Domain-specific systems-on-chip, a class of heterogeneous many-core systems, are recognized as a key approach to narrow down the performance and energy-efficiency gap between custom hardware accelerators and programmable processors. Reaching the full potential of these architectures depends critically on optimally scheduling the applications to available resources at runtime. Existing optimization-based techniques cannot achieve this objective at runtime due to the combinatorial nature of the task scheduling problem. As the main theoretical contribution, this paper poses scheduling as a classification problem and proposes a hierarchical imitation learning (IL)-based scheduler that learns from an Oracle to maximize the performance of multiple domainspecific applications. Extensive evaluations with six streaming applications from wireless communications and radar domains show that the proposed IL-based scheduler approximates an offline Oracle policy with more than 99% accuracy for performance-and energy-based optimization objectives. Furthermore, it achieves almost identical performance to the Oracle with a low runtime overhead and successfully adapts to new applications, many-core system configurations, and runtime variations in application characteristics.