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The process of screening molecules for desirable properties is a key step in several applications, ranging from drug discovery to material design. During the process of drug discovery specifically, protein-ligand docking, or chemical docking, is a standard in-silico scoring technique that estimates the binding affinity of molecules with a specific protein target. Recently, however, as the number of virtual molecules available to test has rapidly grown, these classical docking algorithms have created a significant computational bottleneck. We address this problem by introducing Deep Surrogate Docking (DSD), a framework that applies deep learning-based surrogate modeling to accelerate the docking process substantially. DSD can be interpreted as a formalism of several earlier surrogate prefiltering techniques, adding novel metrics and practical training practices. Specifically, we show that graph neural networks (GNNs) can serve as fast and accurate estimators of classical docking algorithms. Additionally, we introduce FiLMv2, a novel GNN architecture which we show outperforms existing state-of-the-art GNN architectures, attaining more accurate and stable performance by allowing the model to filter out irrelevant information from data more efficiently. Through extensive experimentation and analysis, we show that the DSD workflow combined with the FiLMv2 architecture provides a 9.496x speedup in molecule screening with a <3% recall error rate on an example docking task. Our open-source code is available at https://github.com/ryienh/graph-dock.