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Radio-Frequency Quadrupoles (RFQs) are multi-purpose linear particle accelerators that simultaneously bunch and accelerate charged particle beams. They are ubiquitous in accelerator physics, especially as injectors to higher-energy machines, owing to their impressive efficiency. The design and optimization of these devices can be lengthy due to the need to repeatedly perform high-fidelity simulations. Several recent papers have demonstrated that machine learning can be used to build surrogate models (fast-executing replacements of computationally costly beam simulations) for order-of-magnitude computing time speedups. However, while these pilot studies are encouraging, there is room to improve their predictive accuracy. Particularly, beam summary statistics such as emittances (an important figure of merit in particle accelerator physics) have historically been challenging to predict. For the first time, we present a surrogate model trained on 200,000 samples that yields <6% mean average percent error for the predictions of all relevant beam output parameters, solving the problem of poor emittance predictions by identifying and including hidden variables which were not accounted for previously. These surrogate models were made possible by using the Julia language and GPU computing; we briefly discuss both. We demonstrate the utility of surrogate modeling by performing a multi-objective optimization using our best model as a callback in the objective function to select an optimal RFQ design. We consider trade-offs in RFQ performance for various choices of Pareto-optimal design variables: common issues for any multi-objective optimization scheme. Lastly, we make recommendations for input data preparation, selection, and neural network architectures that pave the way for future development of production-capable surrogate models for RFQs and other particle accelerators.