Deep-learning-based methods for various applications have been shown vulnerable to adversarial examples. Here we address the use of deep-learning networks as inverse problem solvers, which has generated much excitement and even adoption efforts by the main equipment vendors for medical imaging including computed tomography (CT) and MRI. However, the recent demonstration that such networks suffer from a similar vulnerability to adversarial attacks potentially undermines their future. We propose to modify the training strategy of end-to-end deep-learning-based inverse problem solvers to improve robustness. To this end, we introduce an auxiliary net-work to generate adversarial examples, which is used in a min-max formulation to build robust image reconstruction networks. Theoretically, we argue that for such inverse problem solvers, one should analyze and study the effect of adversaries in the measurement-space, instead of in the signal-space used in previous work. We show for a linear reconstruction scheme that our min-max formulation results in a singular-value filter regularized solution, which suppresses the effect of adversarial examples. Numerical experiments using the proposed min-max scheme confirm convergence to this solution. We complement the theory by experiments on non-linear Compressive Sensing(CS) reconstruction by a deep neural network on two standard datasets, and, using anonymized clinical data, on a state-of-the-art published algorithm for low-dose x-ray CT reconstruction. We show a significant improvement in robustness over other methods for deep network-based reconstruction, by using the proposed approach.