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In recent developments in scientific machine learning (SciML), neural surrogate solvers for partial differential equations (PDEs) have become powerful tools for accelerating scientific computation for various science and engineering applications. However, training neural PDE solvers often demands a large amount of high-fidelity PDE simulation data, which are expensive to generate. Active learning (AL) offers a promising solution by adaptively selecting training data from the PDE settings--including parameters, initial and boundary conditions--that are expected to be most informative to help reduce this data burden. In this work, we introduce PaPQS, a Plug-and-Play Query Synthesis AL framework that synthesizes informative PDE settings directly in the continuous design space. PaPQS optimizes the Expected Information Gain (EIG) while encouraging batch diversity, enabling model-aware exploration of the design space via backpropagation through the neural PDE solution trajectories. The framework is applicable to general PDE systems and surrogate architectures, and can be seamlessly integrated with existing AL strategies. Extensive experiments across different PDE systems demonstrate that our AL framework, PaPQS, consistently improves sample efficiency over existing AL baselines.