We investigate the topics of sensitivity and robustness in feedforward and convolutional neural networks. Combining energy landscape techniques developed in computational chemistry with tools drawn from formal methods, we produce empirical evidence indicating that networks corresponding to lower-lying minima in the optimization landscape of the learning objective tend to be more robust. The robustness estimate used is the inverse of a proposed sensitivity measure, which we define as the volume of an over-approximation of the reachable set of network outputs under all additive $l_{\infty}$-bounded perturbations on the input data. We present a novel loss function which includes a sensitivity term in addition to the traditional task-oriented and regularization terms. In our experiments on standard machine learning and computer vision datasets, we show that the proposed loss function leads to networks which reliably optimize the robustness measure as well as other related metrics of adversarial robustness without significant degradation in the classification error. Experimental results indicate that the proposed method outperforms state-of-the-art sensitivity-based learning approaches with regards to robustness to adversarial attacks. We also show that although the introduced framework does not explicitly enforce an adversarial loss, it achieves competitive overall performance relative to methods that do.