Novelty search and related diversity-driven algorithms provide a promising approach to overcoming deception in complex domains. The behavior characterization (BC) is a critical choice in the application of such algorithms. The BC maps each evaluated individual to a behavior, i.e., some vector representation of what the individual is or does during evaluation. Search is then driven towards diversity in a metric space of these behaviors. BCs are built from hand-designed features that are limited by human expertise, or upon generic descriptors that cannot exploit domain nuance. The main contribution of this paper is an approach that addresses these shortcomings. Generic behaviors are recorded from evolution on several training tasks, and a new BC is learned from them that funnels evolution towards successful behaviors on any further tasks drawn from the domain. This approach is tested in increasingly complex simulated maze-solving domains, where it outperforms both hand-coded and generic BCs, in addition to outperforming objective-based search. The conclusion is that adaptive BCs can improve search in many-task domains with little human expertise.