  Deep equilibrium (DEQ) models replace the multiple-layer stacking of
conventional deep networks with a fixed-point iteration of a single-layer
transformation. Having been demonstrated to be competitive in a variety of
real-world scenarios, the adversarial robustness of general DEQs becomes
increasingly crucial for their reliable deployment. Existing works improve the
robustness of general DEQ models with the widely-used adversarial training (AT)
framework, but they fail to exploit the structural uniquenesses of DEQ models.
To this end, we interpret DEQs through the lens of neural dynamics and find
that AT under-regulates intermediate states. Besides, the intermediate states
typically provide predictions with a high prediction entropy. Informed by the
correlation between the entropy of dynamical systems and their stability
properties, we propose reducing prediction entropy by progressively updating
inputs along the neural dynamics. During AT, we also utilize random
intermediate states to compute the loss function. Our methods regulate the
neural dynamics of DEQ models in this manner. Extensive experiments demonstrate
that our methods substantially increase the robustness of DEQ models and even
outperform the strong deep network baselines.
