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For computational efficiency, surrogate models have been used to emulate mathematical simulators for physical or biological processes. High-speed simulation is crucial for conducting uncertainty quantification (UQ) when the simulation is repeated over many randomly sampled input points (aka, the Monte Carlo method). In some cases, UQ is only feasible with a surrogate model. Recently, Deep Neural Network (DNN) surrogate models have gained popularity for their hard-to-match emulation accuracy. However, it is well-known that DNN is prone to errors when input data are perturbed in particular ways, the very motivation for adversarial training. In the usage scenario of surrogate models, the concern is less of a deliberate attack but more of the high sensitivity of the DNN's accuracy to input directions, an issue largely ignored by researchers using emulation models. In this paper, we show the severity of this issue through empirical studies and hypothesis testing. Furthermore, we adopt methods in adversarial training to enhance the robustness of DNN surrogate models. Experiments demonstrate that our approaches significantly improve the robustness of the surrogate models without compromising emulation accuracy.