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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 domain-specific 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.