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While machine learning (ML) has shown increasing effectiveness in optimizing materials properties under known physics, its application in challenging conventional wisdom and discovering new physics still remains challenging due to its interpolative nature. In this work, we demonstrate the potential of using ML for such applications by implementing an adaptive ML-accelerated search process that can discover unexpected lattice thermal conductivity ($κ_l$) enhancement instead of reduction in aperiodic superlattices (SLs) as compared to periodic superlattices. We use non-equilibrium molecular dynamics (NEMD) simulations for high-fidelity calculations of $κ_l$ for a small fraction of SLs in the search space, along with a convolutional neural network (CNN) which can rapidly predict $κ_l$ for a large number of structures. To ensure accurate prediction by the CNN for the target unknown structures, we iteratively identify aperiodic SLs containing structural features which lead to locally enhanced thermal transport, and include them as additional training data for the CNN in each iteration. As a result, our CNN can accurately predict the high $κ_l$ of aperiodic SLs that are absent from the initial training dataset, which allows us to identify the previously unseen exceptional structures. The identified RML structures exhibit increased coherent phonon contribution to thermal conductivity owing to the presence of closely spaced interfaces. Our work describes a general purpose machine learning approach for identifying low-probability-of-occurrence exceptional solutions within an extremely large subspace and discovering the underlying physics.